Experimental and Numerical Study of Laser-Plasma Electron ...

Experimental and Numerical Study of Laser-Plasma Electron Accelerators

Ricardo Parreira de Azambuja Fonseca (Licenciado)

Disserta¸caõ para obten¸cão do grau de Doutor em F´ısica

Outubro de 2002

T´ıtulo: Experimental and Numerical Study of Laser-Plasma Electron Accelerators Nome: Ricardo Parreira de Azambuja Fonseca Doutoramento em F´ısica Orientador: Prof. José Tito da Luz Mendon¸ca Provas conclu´ıdas em: 4 de Outubro de 2002 Resumo O trabalho apresentado nesta disserta¸caõ descreve o estudo experimental e numérico de acelera¸caõ de electrões por interaçcão laser-plasma que decorreu ao longo dos u ´ltimos cinco anos no Grupo de Lasers e Plasmas (GoLP) do Instituto Superior Técnico. Este trabalho é o resultado da colabora¸caõ entre este grupo, o Rutherford Appleton Laboratory (Chilton, Reino Unido) e o Plasma Simulation Group, da Universidade da Califórnia – Los Angeles (Califórnia, E.U.A.). No aˆmbito deste trabalho foi constru´ıdo um espectrómetro de banda larga de electrões para a utiliza¸caõ em experiências de aceleradores por interaçcaõ laser-plasma e são apresentados o desenvolvimento e implementa¸caõ do mesmo. Este espectrómetro permite a caracteriza¸caõ duma banda de energias com a razão 3.2:1 e uma energia máxima de 230 MeV. Apresenta-se também o trabalho desenvolvido no campo da simula¸caõ numérica de aceleradores por interaçcaõ laser-plasma, nomeadamente do “laser wakefield accelerator” (LWFA), do “laser wakefield accelerator” num canal pré-formado e do “selfmodulated laser wakefield accelerator” (SMLWFA) num canal pré-formado. No decorrer deste trabalho foi ainda desenvolvido um cluster de computadores para computa¸caõ avan¸cada que, tanto quanto sabemos, é o sistema de computa¸caõ mais possante existente em Portugal. Palavras-chave: • Interaçcão laser-plasma • Aceleradores a Plasma • Espectrómetro de Electrões • Simula¸cão Numérica de Plasmas • Lasers de Alta Potência • Clusters de Computadores

Title: Experimental and Numerical study of Laser-Plasma Electron Accelerators

Abstract

The work presented in this thesis describes the experimental and numerical study of laser-plasma electron accelerators done over the last five years at the Grupo de Lasers e Plasma (GoLP) of the Instituto Superior Técnico (IST). This work is the result of a collaboration between this group, the Rutherford Appleton Laboratory (RAL, UK) and the Plasma Simulation Group from the University of California - Los Angeles, (California, USA). In the scope of this project we have built a broad range electron spectrometer for the study laser-plasma electron accelerators and we present here the development and implementation of this system. This spectrometer allows for the characterization of a 3.2:1 energy range and a maximum energy of 230 MeV. We also present the work done on the numerical simulation of laser-plasma electron accelerators, namely the laser wakefield accelerator (LWFA), the channeled laser wakefield accelerator and the channeled self-modulated laser wakefield accelerator. During this work a computer cluster for numerical computational was also developed which is, to our knowledge, the most powerful machine for numerical computation in Portugal. Keywords: • Laser-Plasma Interaction • Plasma Based Accelerators • Electron Spectrometer • Numerical Simulation of Plasmas • High Power Lasers • Computer Clusters

Acknowledgements This thesis is the result of a collaboration between the Grupo de Lasers e Plasmas at the Instituto Superior Técnico of the Universidade Técnica de Lisboa, Portugal, the Plasma Simulation Group at University of California Los Angeles, USA, and the Rutherford Appleton Laboratory, Chilton, UK. Many have contributed to it, directly and indirectly, and I feel indebted for their collaboration and support, without which this work would not have been possible. In particular I would like to thank: First of all, my supervisor, Prof. J. T. Mendon¸ca, for guiding me through this project, and providing me the conditions for this work. His contribution is immense and I am extremelly grateful to him for all I’ve learned in the past five years; I would especially like to thank Prof. Luis Silva for all the support and opportunities he made available for me, and for his invaluable help and collaboration that have changed my perspectives in physics; From the UCLA Plasma Simulation Group, Prof. Warren Mori and the late Prof. John M. Dawson for inviting me to work in their group, and for their guidance, inspiration, and support; Dr. Viktor Decyk, for kindly sharing with me small portions of his omniscient knowledge on Fortran and parallel computing; Dr. C. Clayton for helpful discussions regarding the electron spectrometer; Everyone else at the UCLA group, especially Dean Dauger, Evan Dodd, Brian Duda, Roy Hemker, Chuang Ren, John Tonge, and Frank Tsung, and also Seung Lee from USC; From RAL, Prof. R. Bingham for his continued support and for making available the know-how and materials for the development of the detection system and to Dr. Barry Kellet for his collaboration and support in the development and construction of the detection system; The GoLP researchers who help me directly on this project namely N. Lopes, for his help in the integration of the electron spectrometer and overall electromechanical work, Dr. J. M. Dias, for his help with LabView, Dr. G. Figueira

vi

Acknowledgements for the help with the pagination of the Thesis, M. Eloy, that worked in the beginning of the electron spectrometer project with me, and G. Sorasio, for lowering the volume on his Italian radio stations when I was at critical stages of the writing of this thesis; Eng. Pedro Machado, for helpful discussions on computing science and technology and Dr. J. Barbosa, for his invaluable help with the GEANT code; Everyone else at GoLP that share the day-to-day experience of a research group with me namely Cristina Carias, Luis Cardoso, Isabel Carvalho, Hélder Crespo, Dr. J. Davies, Dr. M. Fajardo, Elsa Fonseca, Ariel Guerreiro, Michael Marti, José Rodrigues, Carla Rosa, and Carlos Russo, and Prof. David Resendes and Prof. Ana Maria Martins, and our administrative assistants, Anabela Gon¸calves and Ilda Caetano; To the Funda¸caõ para a Ciência e Tecnologia, Funda¸caõ Gulbenkian e Funda¸caõ Luso-Americana para o Desenvolvimento without whose financial support this work would never have taken place; And to all the members of the Jury. Gostaria ainda de agradecer em Portugês: ` malta das sextas que, entre copos de vinho, atura as minhas deambula¸co˜es A incoerentes e me mantém mais ou menos racional; Aos meus pais, por todo apoio que sempre me deram nesta estranha op¸cão; E a ti Rute, que me ofereces o repouso do teu corpo, por existires e me deixares encontrar-te, e me dares tudo sem nada pedir.

Contents Introduction

1

1 Laser-Plasma Accelerator Concepts

5

1.1

Plasma Wave Generation . . . . . . . . . . . . . . . . . . . . . .

5

1.2

Nonlinear Plasma Waves . . . . . . . . . . . . . . . . . . . . . .

7

1.3

Electron acceleration and Detuning . . . . . . . . . . . . . . . . 11

1.4

Laser Propagation in Plasmas . . . . . . . . . . . . . . . . . . . 13

1.5

1.4.1

Guiding . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4.2

Laser-Plasma Instabilities . . . . . . . . . . . . . . . . . 17

Laser-Plasma Electron Accelerators . . . . . . . . . . . . . . . . 19 1.5.1

Laser Wakefield Accelerator . . . . . . . . . . . . . . . . 20

1.5.2

Self-Modulated Laser Wakefield Accelerator . . . . . . . 23

2 Spectrometer Design 2.1

2.2

2.3

27

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.1.1

Sector Field Lenses . . . . . . . . . . . . . . . . . . . . . 28

2.1.2

Particle Transport Systems . . . . . . . . . . . . . . . . . 36

Broad Range Electron Spectrometers . . . . . . . . . . . . . . . 37 2.2.1

Circular Shaped Magnetic Fields . . . . . . . . . . . . . 38

2.2.2

Numerical Simulation for Circular Fields . . . . . . . . . 42

Dimensioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

viii

Contents 3 Detection System 3.1

55

Charged Particles Detection . . . . . . . . . . . . . . . . . . . . 55 3.1.1

Coulomb Collisions . . . . . . . . . . . . . . . . . . . . . 56

3.1.2

Energy Loss for Electrons . . . . . . . . . . . . . . . . . 58

3.1.3

Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . 59

3.2

Scintillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.3

Semiconductor Diode Detectors . . . . . . . . . . . . . . . . . . 67

3.4

Electron Detectors Used . . . . . . . . . . . . . . . . . . . . . . 68 3.4.1

Scintillating Plate . . . . . . . . . . . . . . . . . . . . . . 69

3.4.2

CsI(Tl) Scintillators

. . . . . . . . . . . . . . . . . . . . 70

4 Electron Spectrometer System 4.1

73

Deflecting element . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.1.1

Magnet Stand . . . . . . . . . . . . . . . . . . . . . . . . 74

4.1.2

Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2

Vacuum box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3

Detection System . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.4

Computer Interface . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 PIC Simulation of Plasmas

99

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.2

Numerical Simulation of Plasmas . . . . . . . . . . . . . . . . . 99

5.3

PIC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4

Osiris

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.4.1

Development . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4.2

Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4.3

OSIRIS Framework . . . . . . . . . . . . . . . . . . . . . 110

5.4.4

Performance . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.5

Visualization and Data-Analysis Infrastructure . . . . . . . . . . 112

5.6

EP2 Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Contents

ix

6 Simulation of Laser Acceleration Experiments at L2I

119

6.1

The L2I Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2

Laser Wakefield Accelerator . . . . . . . . . . . . . . . . . . . . 122

6.3

6.4

6.2.1

Wakefield . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.2.2

Particle Dynamics . . . . . . . . . . . . . . . . . . . . . 125

6.2.3

Energy Distribution . . . . . . . . . . . . . . . . . . . . . 127

Channel Laser Wakefield Accelerator . . . . . . . . . . . . . . . 128 6.3.1

Laser Spot . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.3.2

Wakefield . . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.3.3

Time Evolution of the Accelerating Wake . . . . . . . . . 133

6.3.4

Wakefield Phase Velocity . . . . . . . . . . . . . . . . . . 145

6.3.5

Wakefield wavelength . . . . . . . . . . . . . . . . . . . . 146

6.3.6

Maximum Accelerating Gradient . . . . . . . . . . . . . 146

6.3.7

Particle energy . . . . . . . . . . . . . . . . . . . . . . . 148

6.3.8

Ejected bunch . . . . . . . . . . . . . . . . . . . . . . . . 152

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7 Overview and Future Work

157

Bibliography

160

List of Figures 1.1

Plasma wave produced in the laser wakefield accelerator regime in 1D, with a laser pulse with duration L = λp = 45 µm for (a) a0 = 0.5 and (b) a0 = 2.0. Simulations were done using the OSIRIS code (see Chapter 5 for details). . . . . . . . . . . . . . 10

1.2

Wakefield (a) and Plasma wave density (b) in the laser wakefield accelerator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3

Momentum vs. Position in the laser wakefield accelerator.

1.4

Laser field envelope in the self-modulated laser wakefield accelerator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.5

Plasma wave density (a) and Momentum vs. Position (b) in the self-modulated laser wakefield accelerator after 2.1 ps of laser propagation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1

Focusing of charged particles by a homogeneous magnetic sector field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2

Equivalent system using a thin lens. Note that in this coordinate system beam deflexion is not considered. . . . . . . . . . . . . . 31

2.3

The defocusing effect caused by oblique entrance and exit angles. A beam originating from A’ is focused in B’. For perpendicular entrance and exit angles the image-object relation would be between A and B. Note the definition of 0 and/or 00 . . . . . 31

2.4

Beam trajectories on a sector field for arbitrary deflection angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5

Focal distance for a sector field as a function of the deflection angle, for the following parameters: a0 = 19.9 cm, θ0 = 10o and θ00 = 10o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

. . . 22

xii

List of Figures 2.6

Focal length of a sector field as a function of the deflection angle for the following parameters: a0 = 19.9 cm, θ0 = 10o and θ00 = 10o . 35

2.7

Fringe Field. The real magnetic field from a dipole magnet is presented in comparison with the idealized magnetic field. The dashed line represents the virtual field boundary. . . . . . . . . 35

2.8

Triple focusing dipole system. . . . . . . . . . . . . . . . . . . . 37

2.9

Circular shaped magnetic field. For each deflection angle α, this system behaves as a sector field with the same angle. . . . . . . 39

2.10 Circular shaped magnetic field with an entry angle. This angle provides some vertical focusing. . . . . . . . . . . . . . . . . . . 41 2.11 Defocussing lens effect. The object appears to be closer than it really is because of the increase in beam divergence. . . . . . . . 41 2.12 Fringe field calculated from (2.24) for the two situations presented. Note that the horizontal axis is in units of D, the air gap distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.13 Complete magnetic field for the modeling of the magnet. The entry angle is 10o and the field radius is 11.5 cm. The blue solid lines represent the physical boundaries of the poles pieces. . . . 44 2.14 Coordinate system for the calculations of the fringing field horizontal components. . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.15 Horizontal Fringe Field. (a) and (b) show the X and Y components of the fringing field for a vertical coordinate of 0.02 D normalized to the magnetic flux in the uniform field region. The dimensions for the magnet are the same as in figure 2.13. . . . . 45 2.16 Deflection angle for electrons for magnetic fields from 0.5 T to 3 T. The magnet has a radius of 11.5 cm, and an air gap of 1.3 cm. Calculations were done using the long-tail model. The solid line shows the analytical solutions, and the symbols show the numerical solutions. . . . . . . . . . . . . . . . . . . . . . . 46 2.17 Focal lines for a circular magnet with an entrance angle. The magnet has a radius of 11.5 cm, an entry angle of 10o and an air gap of 1.3 cm. The particles originate from a point 85 cm away from the magnets entry angle. The solid line shows the analytical solutions, and the symbols show the numerical solutions. Calculations were made for three fringe field models: ideal, short-tail and long-tail. On this scale the differences between short and long tail are not visible. . . . . . . . . . . . . . 47

List of Figures

xiii

2.18 Lateral Magnification as a function of the deflection angle. The magnet has a radius of 11.5 cm, an entry angle of 10o and an air gap of 1.3 cm. The particles originate from a point 85 cm away from the magnets entry angle. The solid line shows the analytical results and circles show the numerical results. . . . . 49 2.19 Longitudinal aberration for a system with a 11.5 cm radius, an entry angle of 10o and an air gap of 1.3 cm. . . . . . . . . . . . 50 2.20 Deflection angle for 200 MeV electrons as function of the magnet radius ρ0 for several values of the magnetic field. The source point is placed 1 m away from the entrance of the magnet. . . . 51 2.21 Focal point distance to the center of the magnet (q) as a function of the deflection angle for several values of the magnetic field radius ρ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.22 Final pole piece design for our electron spectrometer. . . . . . . 53 3.1

Total, collisional and radiative stopping power for fast electrons for Helium and Aluminum. The dotted red and blue lines show the collisional and radiative energy loss. The solid black line shows the total stopping power. Note that the stopping power is normalized to the the density, ρ. . . . . . . . . . . . . . . . . 61

3.2

Total stopping power for Mylar, Air and Helium. . . . . . . . . 62

3.3

CSDA Range, R, for Mylar, Air and Helium.

3.4

CURIX ORTHO REGULAR screens from Agfa used for electron detection and counting. . . . . . . . . . . . . . . . . . . . . 70

3.5

CsI crystals mounted on photodiodes. The left detector is presented without the protective wrapping for clarity. . . . . . . . . 71

4.1

The deflection magnet for the electron spectrometer system placed in front of the main L2I vacuum chamber (a) and a detail of the pole piece (b). Note that for (b) we separated the two magnet halves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2

The BLMPS MON 1 16/500 C4 power supply (foreground) for the deflecting magnet (background). . . . . . . . . . . . . . . . . 76

4.3

3D rendering of the magnet stand, showing the alingment and positioning elements (a) and the actual stand with the deflecting magnet on top (b). . . . . . . . . . . . . . . . . . . . . . . . . . 77

. . . . . . . . . . 63

xiv

List of Figures 4.4

Magnetic field as a function of supply current. . . . . . . . . . . 78

4.5

Magnetic flux density near the entry edge of the pole piece for three field intensities. . . . . . . . . . . . . . . . . . . . . . . . . 79

4.6

Magnetic flux density near the exit edge of the pole piece for three field intensities for 58o (a) and 120o (b) deflection angles. . 80

4.7

Fitting curves for the distribution of the magnetic field. . . . . . 81

4.8

Simulation field from measured values, sample trajectories and focal points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.9

Deflection angle as a function of χ = B/p. . . . . . . . . . . . . 83

4.10 Effective Field Boundary Calculation. Simulation results shown as black circles and numerical fit as a red line. Calculations were done for B = 50%. . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.11 Effective Field Boundary Calculation for the simplified parameter χs . Simulation results shown as black circles and numerical fit as a red line. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.12 Focus point distance to the center of the magnet. Simulation results shown as black circles and the EFB calculations as a red line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.13 Depth of focus for the magnetic system. The particle beam as a spot size below S × W between the two corresponding lines. . 85 4.14 GEANT4 simulation for a) 10 MeV, b) 20 MeV, c) 50 MeV, d) 100 MeV, e) 200 MeV and f) 500 MeV electrons crossing a 0.1 mm Mylar window and propagating for 1.3 m in air. The total simulation box is 0.5 × 0.5 × 2.0m3 . . . . . . . . . . . . . . 87 4.15 GEANT4 simulation for 10 MeV electrons with several window/world materials: a) Mylar/Air, b) Mylar/Helium, c) Vacuum/Air, d) Vacuum/Helium, and e) Mylar/Vacuum. The remaining simulation parameters are the same as in fig. 4.14. . . . 89 4.16 Vacuum box for the spectrometer. The top half of the magnet is not shown for clarity. . . . . . . . . . . . . . . . . . . . . . . . 90 4.17 Signal processing circuit for the data acquisition system. Note: The actual AmpOps used were the LMC2001 which have the same pinout as the LMC7101 but much better performance. . . . 92 4.18 Signal processing waveforms for the data acquisition system. Waveforms shown are from SPICE simulations. . . . . . . . . . 93

List of Figures

xv

4.19 Detector electronics prototype. . . . . . . . . . . . . . . . . . . . 94 4.20 Trigger circuit for testing and calibration.

. . . . . . . . . . . . 95

4.21 Data acquisition and Computer Interface circuit. . . . . . . . . . 96 4.22 Data acquisition and computer interface prototype. . . . . . . . 97 5.1

Loop cycle for the electromagnetic PIC algorithm used. The particles are numbered i = 1, 2, · · · , NP and the grid indexes are j, which becomes a vector in 2 and 3 dimensions. . . . . . . 104

5.2

Osiris main class hierarchy. . . . . . . . . . . . . . . . . . . . . . 108

5.3

A typical cycle, one time step, in an OSIRIS 2 node run. The arrows show the direction of communication between nodes. . . 110

5.4

Force field acting on the 30 GeV SLAC beam inside a plasma column. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.5

The EP2 Cluster at IST. The cluster has 16 dual Macintosh G4’s, an Asanté Intracore 8000 10/100 network switch (top left) and two 17” Monitors for maintenance and installation purposes.115

5.6

Network performance of the EP2 cluster with packet sizes ranging from 1 byte to 2 Megabytes, using half-duplex (Ping) and full-duplex (Swap) communications. . . . . . . . . . . . . . . . . 116

5.7

Parallel performance of OSIRIS 2D running on the EP2 cluster for the granularity benchmarks. . . . . . . . . . . . . . . . . . . 117

5.8

Parallel performance of OSIRIS 2D running on the EP2 cluster for the scalability benchmarks, normalized to the 1 node run. Note that the vertical axes scales from 90% to 100%. . . . . . . 118

6.1

General view of the L2I laboratory showing the laser system in the foreground and the interaction area in the background. . . . 120

6.2

Shadowgraphy image of a laser triggered discharge plasma channel produced at the L2I laboratory. Courtesy of N. C. Lopes. . . 122

6.3

Mass Density (a) and Accelerating Field (b) for the laser wakefield in the laser wakefield accelerator run. . . . . . . . . . . . . 124

6.4

Phasespace density (p1-x1) for the background electrons showing particle trapping and acceleration. . . . . . . . . . . . . . . 126

6.5

Phasespace plot (p1-x1) for the two test species: particles injected with γ = 10 (black) and particles injected with γ = 50 (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

xvi

List of Figures 6.6

Electron density along the x1 direction for the two test species: particles injected with γ = 10 (black) and particles injected with γ = 50 (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.7

Energy density for the two test species, normalized to the maximum density: particles injected with γ = 10 (black) and particles injected with γ = 50 (red). . . . . . . . . . . . . . . . . . . 128

6.8

Channel density profile used in the simulations. . . . . . . . . . 130

6.9

Laser spot size evolution. (a) Shows a comparison between uniform plasma propagation and channel propagation and (b) compares the multiple channel regimes simulated. . . . . . . . . . . 132

6.10 Density Wake (a) and Accelerating Field (b) for the τL = 0.5λp run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.11 Density Wake (a) and Accelerating Field (b) for the τL = 1λp run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.12 Density Wake (a) and Accelerating Field (b) for the τL = 2λp run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.13 Density Wake (a) and Accelerating Field (b) for the τL = 3λp run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.14 Density Wake (a) and Accelerating Field (b) for the τL = 5λp run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.15 Density Wake (a) and Accelerating Field (b) for the τL = 10λp run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.16 Density Wake (a) and Accelerating Field (b) for the τL = 20λp run. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.17 Time evolution of the accelerating wake for the τL = 0.5λp run. . 141 6.18 Time evolution of the accelerating wake for the τL = 1λp run. . . 142 6.19 Time evolution of the accelerating wake for the τL = 2λp run. . . 142 6.20 Time evolution of the accelerating wake for the τL = 3λp run. . . 143 6.21 Time evolution of the accelerating wake for the τL = 5λp run. . . 143 6.22 Time evolution of the accelerating wake for the τL = 10λp run. . 144 6.23 Time evolution of the accelerating wake for the τL = 20λp run. . 144

List of Figures 6.24 Wakefield phase velocity (γp ) for all the regimes simulated. The measured velocities are marked as circles and the solid lines show several theoretical models: (a) is the 1D limit, (b) includes the 2D (curvature) corrections, (c) is the 1D limit using the corrected density and (d) is the same as (c) using the corrected density. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.25 Wakefield wavelength in the bottom of the channel (λp0 ) for all the regimes simulated. The measured wavelengths are marked as circles and the solid lines show several theoretical models: (a) is the 1D limit, (b) is the 1D limit using the corrected density as determined for the wakefield phase velocity and (c) is the same as (b) using the nonlinear correction. . . . . . . . . . . . . . . . 147 6.26 Maximum accelerating gradient for all the regimes simulated. The measured accelerating gradients are marked as circles and the solid lines show several theoretical models: (a) is the nonrelativistic wavebreaking limit, (b) is the relativistic wavebreaking limit, (c) is the same as (a) using the corrected density and (d) is the same as (b) using the corrected density and the corrected γp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.27 Evolution of the maximum energy of the background plasma (a) and the injected particles (b). . . . . . . . . . . . . . . . . . . . 150 6.28 Energy distribution for the τL = 2λp run, for t ' 3.3ps for the background plasma (a) and the injected particles (b). . . . . . . 151 6.29 Section of the mass density for the τL = 5λp run, for t ' 3.3 ps showing the ejected particle bunch (dotted circle). . . . . . . . 153 6.30 Section of the p1x1 phasespace density for the τL = 5λp run, for t ' 3.3 ps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.31 Plot of the p2p1 phasespace density for the τL = 5λp run, for t ' 3.3ps showing the ejected particle bunch (dotted circle). . . 154

xvii

List of Tables 1.1

Laser plasma accelerator parameters for several plasma densities. Calculations were done using a λ = 1.053 µm laser focused to a 10 µm spot size diameter. The 3D values include the wavefront curvature correction. . . . . . . . . . . . . . . . . . . . . . 12

2.1

Definitions for Short-Tail e Long-Tail fringing fields. . . . . . . . 43

2.2

Magnet Parameters chosen. . . . . . . . . . . . . . . . . . . . . 53

3.1

Density (ρ), Mean Excitation Potential (I), and Effective Atomic Number (Z) for several absorbing materials. . . . . . . . . . . . 61

3.2

Total stopping power for several absorbing materials, for electron energies from 0.1 MeV to 1 GeV. . . . . . . . . . . . . . . . 63

3.3

CSDA range, R, in centimeters, for several absorbing materials, for electron energies from 1 MeV to 1 GeV. . . . . . . . . . . . . 64

3.4

Physical properties of common Scintillators. . . . . . . . . . . . 65

3.5

Average energy required for electron-hole pair creation in silicon and germanium. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1

Fitting parameters for the measured magnetic field. . . . . . . . 79

4.2

Effective field boundary. Values are in cm. . . . . . . . . . . . . 82

4.3

Simulation results for GEANT4 runs. . . . . . . . . . . . . . . . 88

5.1

Typical push time for three machines, in two and three dimensions. Values are in µs/particle × node. . . . . . . . . . . . . . . 112

5.2

EP2 Cluster characteristics. . . . . . . . . . . . . . . . . . . . . 114

xx

List of Tables 6.1

Current L2I laser paramters. a0 , w0 , and I values are shown for the 30 cm parabolic mirror configuration. . . . . . . . . . . . . . 121

6.2

Simulation parameters for the laser wakefield accelerator run. . . 123

6.3

Simulation parameters common to all the channel electron accelerator runs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.4

Simulation parameters used in each of the channel electron accelerator runs. The rightmost column shows the ration between the propagation length simulated and the corresponding dephasing length, Ld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Introduction In the last 70 years particle accelerators have become important tools to experimentally study the basic constituents of matter and the fundamental laws of nature. Many important discoveries revolutionized our understanding of the nature of the universe and spawned many technological applications. Though tremendous progress has been made, our present theory of the physical world is not complete. In order to experimentally pursue the quest for the grand unified theory ever more powerful accelerators are needed. Present accelerators explore particle masses of up to about 150 GeV. A new particle is produced, for example, if an electron and positron, both with an accelerated energy of 45.6 GeV collide and convert (annihilate) into a Z-Boson with a mass of 91.2 GeV. In order to advance science into unknown regions beyond our present knowledge, accelerators with a larger final beam energy are required. The maximum accelerating gradient being achieved in RF cavities of existing facilities are of the order of tens of MV/m. To reach the required energies we thus resort to very large accelerators. The Stanford Linear Accelerator Center (SLAC) accelerator is roughly 3 Km long and achieves about 50 MeV using an 25 MV/m accelerating gradient , and the Large Electron Positron collider (LEP) at CERN is a 27 Km long accelerating ring, achieving energies of about 100 GeV using 7 MV/m accelerating gradients. To achieve higher energies, we would need to either increase the accelerating gradient or to increase the accelerator sizes. The latter is obviously limited by technological and, more importantly, economical reasons. The first is also limited by physical and technological problems. The accelerating gradients are presently limited to roughly 100 MV/m mostly because of the breakdown of the walls of the accelerating structure trough tunnel ionization. As present accelerator technologies start to reach their limits, other options are beginning to be explored. Plasma-based accelerators are of great

2

Introduction interest because of their ability to sustain extremely large gradients and are therefore a means of overcoming the breakdown constraint. Ionized plasmas can sustain electron plasma waves with electric fields on the order of the nonrelativistic wavebreaking field: 1/2

E0 [V/cm] ' 0.96n0 [cm−3 ]

(1)

where n0 is the background electron density. For a typical laser produced laboratory plasma of n0 = 1019 cm−3 this gives E0 ' 300GV/m which is over three orders of magnitude greater than the value obtained in conventional RF cavities. Tajima and Dawson [1] first introduced the concept of laser electron accelerators in 1979 and since then tremendous progress has occurred in this field. This is partly due to the advance in laser technology, particularly the development of compact, multi-terawatt laser systems based on chirped pulse amplification, such as the laser system at L2I/GoLP. There has been experimental evidence of accelerating gradients in excess of 100 GV/m, and novel acceleration schemes are proposed on a regular basis. These laser driven, compact laser accelerators can overcome some of the limitations of conventional accelerators, and are likely to play an important role in a new generation of particle accelerators that will support further understanding of the structure of matter, the fundamental forces of nature and the origin of the universe. In this thesis, we present the work done during the past five years in the experimental and numerical study of electron accelerators in connection to the work being done at the Grupo de Lasers e Plasmas in the Instituto Superior Técnico of the Universidade Técnica de Lisboa, Portugal. A great deal of effort was given to the acquisition of the required know-how that was absent in our group and the total work done in this period often exceeded the scope of this thesis, and overlapped with adjoining fields, like astrophysics, and inertial confinement fusion. Extensive work was also done on short laser pulse diagnostics, visualization and data analysis, and optical ray-tracing for image acquisition systems design. Chapter 1 presents an overview of plasma based accelerator concepts. Special relevance is given to wakefield generation, nonlinear plasma waves, and electron acceleration and detuning. We also approach laser pulse propagation in plasmas, including guiding mechanisms and laser-plasma instabilities. Finally we present an overview of laser-plasma electron accelerators. Chapters 2 to 4 describe the electron spectrometer system built for laserplasma accelerator experiments. Chapter 2 introduces the magnetic dispersion

3 system theory, and describes in detail the dimensioning and construction of the magnet used. Chapter 3 presents an overview of detection systems for these instruments, and describes the dimensioning and construction of the detectors for this spectrometer . Finally, Chapter 4 describes the integration of the whole system with the L2I laboratory, with special detail on the mechanical positioning, detector electronics and vacuum box. Chapters 5 and 6 present the simulation work done in regard to the electron accelerator experiments at the L2I laboratory. Chapter 5 presents a review of Particle-In-Cell algorithms, the OSIRIS framework used for the simulations, and the EP2 cluster where the simulations were run. Chapter 6 presents the simulation work done for acceleration schemes in conditions relevant to the L2I system. Finally, chapter 7 presents a summary of the work presented on this thesis along with future prospects and conclusions.

Chapter 1 Laser-Plasma Accelerator Concepts Tajima and Dawson first suggested the use of lasers to produce high amplitude plasma waves for electron acceleration in 1979 [1]. The general physical principle behind this accelerator is quite simple: if we inject an electron into a high amplitude plasma wave, and if this electron had approximately the same velocity as the wave, it will stay in phase with the electric field, absorbing energy from the wave and accelerating. At the time laser technology had progressed significantly, and Terawatt level laser systems were beginning to appear, making the generation of high amplitude plasma waves through laserplasma interaction a reality. Nowadays, table-top, multi-terawatt laser systems like the L2I laser at the Grupo de Lasers e Plasmas at the Instituto Superior Técnico [2], in Lisboa, Portugal, are available, allowing for the construction of compact electron accelerators. This chapter presents an overview of laser-plasma accelerator concepts. We begin by discussing plasma wave generation through the use of intense laser pulses, and we will then proceed with the analysis of nonlinear plasma waves, and electron acceleration and detuning. The propagation of laser pulses in plasmas will also be analyzed and finally we will look into laser-plasma electron accelerator configurations.

1.1

Plasma Wave Generation

In the discussion of high-intensity laser plasma interaction it is convenient to use the normalized vector potential a = eA/me c2 . We define the laser strength

6

Laser-Plasma Accelerator Concepts parameter, a0 , as the peak amplitude of the normalized vector potential of the laser field. This parameter is related to the laser peak intensity I and power P of a linearly polarized laser pulse by: a0 =

2e2 λ2 I πm2e c5

1/2

' 8.6 × 10−10 λ[µm]I 1/2 [W/cm2 ]

(1.1)

where P [GW] ' 21.5(a0 w0 /λ)2 , w0 is the laser spot radius at focus, λ = 2π/k is the laser wavelength, ω = ck is the laser frequency, and we assume a vector potential in the form a = exp(−ρ2 /w02 ) cos(kz − ωt)ex , where ρ is the radial coordinate. Note that the normalized vector potential a0 corresponds to the normalized transverse quiver velocity, γv⊥ /c, of the electrons in the laser field. In laser-plasma accelerators the wakefield is generated through the ponderomotive force. This force is associated with gradients in the laser field envelope. While an electron oscillates in laser field, if the amplitude of the oscillation is sufficient, it will travel to a region where the envelope amplitude of the field is smaller. If this is the case the field force will be insufficient to restore the electron to its initial position and the center of oscillation will be shifted. A simple derivation of this force can be found in [3]. Consider the electron momentum equation in the cold fluid limit, dp/dt = −e[E + (v × B)], where d/dt is the convective derivative d/dt = ∂/∂t + v · ∇. We can write the electric and magnetic fields as E = −∂A/∂ct and B = ∇ × A. In the linear limit |a| = e|A|/me c2 1, and the first order electron equation of motion is simply me ∂vq /∂t = −eE, which gives a quiver velocity of vq = ca. Assuming v = vq + δv the second order motion is then given by:

dδp/dt = −me [(vq · ∇)vq + cvq × (∇ × a)] = −me c2 ∇(a2 /2) Which means that the three dimensional ponderomotive force in the linear limit (a2 1) is: Fp = −me c2 ∇(a2 /2)

(1.2)

The ponderomtive force can also be viewed as the gradient of the radiation pressure, where regions of higher laser field intensity will push charged particles toward regions of smaller field intensity. We can now imagine, for example, a laser pulse with a duration of 2π/ωp . The leading part of the pulse will push electrons away from the optical axis of propagation while the backward edge will push them back into this axis. Because the pulse has the same

1.2 Nonlinear Plasma Waves

7

length as the plasma wavelength, this force will resonate with the plasma and generate high amplitude waves.

1.2

Nonlinear Plasma Waves

Wakefield analysis has been successfully analyzed in the linear 3-D regime using the cold fluid equations i.e. Poisson’s equation, the continuity equation and the momentum equation. In this regime, the plasma wave generated in an initially uniform plasma by a laser pulse can be described by [4, 5, 6]:

∂2 + ωp2 ∂t2

a2 δn = c2 ∇ 2 n0 2

(1.3)

where ωp is the electron plasma frequency, ωp = (4πn0 e2 /me )1/2 , n0 the background plasma density, δn/n0 1 the normalized amplitude of the plasma wave produced and a2 1 the normalized intensity of the laser pulse. The solution to (1.4) is then: δn c2 = n0 2ωp

Z

t

sin [ωp (t − t0 )] ∇2 a(t0 )2 dt0

(1.4)

0

The electric field of the wake can now be calculated using ∇2 φ = kp2 (δn/n0 ). Using (1.4) we get: me c2 ωp E(r, t) = − 2e

Z

t

sin [ωp (t − t0 )] ∇2 a(r, t0 )2 dt0

(1.5)

0

This result implies that the perpendicular extent of the wake will be of the order of the laser spot size, 2 w0 . We can now find the solutions for a single linearly polarized laser pulse propagating along ez with an intensity profile given by a2 = a20 exp(−2ρ2 /w02 ) sin2 (πξ/L) for 0 < ξ < L, where ρ is again the radial coordinate, ξ = z −ct, and L is the pulse length. The solutions to equation (1.5) show that optimal excitation occurs for when pulse length is near the plasma skin depth, i.e., L ' λp . Trailing the laser pulse we get, for L = λp , in cylindrical coordinates, a wakefield density and electric field of [5, 6]: δn π 2 8 2ρ2 2ρ2 = − a0 1 + 2 2 1 − 2 exp − 2 sin(kp ξ) n0 8 kp w0 w0 w0

(1.6)

8

Laser-Plasma Accelerator Concepts

2ρ2 π 2 Ez = − a0 exp − 2 cos(kp ξ) E0 8 w0

(1.7)

where E0 is again the nonrelativistic wavebreaking field (1.8). In addition to the axial wakefield Ez , transverse wakefields Er and Bθ will also be generated [7, 8]. A relativistic electron being accelerated by a wakefield with a phase velocity also close to the speed of light will experience a radial force proporcional to Er − Bθ , which will be null on the axis. For a particle displaced from the axis there will be regions of focusing and defocusing wakefields, combined with regions of accelerating and decelerating wakefields. It has been shown that there is a phase region of the wake where both accelerating and focusing forces exist [7]. For a high intensity laser driver, a0 & 1 , the linear approach presented above is no longer valid. Some theoretical studies have been done for broad drivers, kp ρ⊥ 1, where ρ⊥ is the transverse dimension of the driver, making this a 1D problem, and assuming that the laser field is nonevolving [9, 10]. However, in 2 and 3 dimensions, detailed understanding of nonlinear plasma waves usually requires numerical simulation. The codes used are usually based on the quasi static approximation in 2D [11, 12], which assumes that the time it takes for the laser pulse to evolve is much larger than electron transit time, and then creates a fluid description based on this assumption, evolving the laser field independently, and particle-in-cell codes in 2D and 3D, which are described in detail in Chapter 5, which present a numerically heavier but more accurate approach to modeling these interactions, especially at high intensities. One key parameter in laser-plasma accelerators is the maximum wave amplitude that the plasma can sustain. This value can be easily estimated through linear theory in 1D assuming that the wakefield has the form Ez = Emax sin[ωp (z/vp − t)], where vp ' c is the plasma wave phase velocity. The peak field amplitude Emax can be determined using Poisson’s equation, ∇·E = 4πeδn, assuming that all electrons in the plasma are oscillating at the plasma frequency ωp , with a wavenumber kp = ωp /c. In this situation the plasma can no longer supply electrons to increase the wave amplitude, like a surface wave on the ocean reaching the beach, which results in wavebreaking. In this situation we have (ωp /c)Emax = 4πen0 , or Emax = E0 , the non-relativistic wavebreaking field [13] given by: E0 =

cme ωp e

(1.8)

For a typical laser produced laboratory plasma of n0 = 1019 cm−3 this gives E0 ' 300GV/m. This wavebreaking limit changes when relativistic and

1.2 Nonlinear Plasma Waves

9

thermal effects are considered. It has been shown [14], using nonlinear relativistic cold fluid equations, that the √ maximum amplitude of an electron plasma wave is increased by a factor of 2(γp − 1)1/2 , where γp is the relativistic factor for the plasma wave, γp = (1 − vp2 /c2 )−1/2 . The relativistic wavebreaking limit is then: EW B =

√ 2(γp − 1)1/2 E0

(1.9)

However, when thermal effects are considered, this enhancement is reduced, because hot electrons from the thermal distribution will become trapped by this plasma wave and forcing wavebreaking to occur sooner. Using warm relativistic fluid theories it has been shown [15] that the thermal wavebreaking field amplitude, Eth , at a given plasma temperature T and for large values of 1/2 γp βth then becomes: −1/4

Eth = βth

q 1/2 1/4 ln(2γp βth )E0

(1.10)

where βth = 3T /me c2 . Assuming a plasma temperature T = 10eV and a wave relativistic factor of γp = 300, we get Eth ' 12E0 . For the same density as the above example this yields Eth ' 3.6TV/m, which is is over four orders of magnitude greater than the value obtained in any conventional RF cavity. Figure 1.1 shows an example of simulated 1D wakefields in the resonant condition described above, for low and high intensity drive beams. The simulation was done with a τL = 150 fs, λ = 1.053µm laser pulse using the OSIRIS framework described in chapter 5. The solid black line shows the electric field and the solid red line the density perturbation δn/n0 . In the low intensity linear regime, we can see that the plasma is simply an oscillation with frequency ωp and a phase velocity determined by the driver velocity, as predicted above. In the high intensity case the plasma wave becomes strongly non-linear. The density oscillations become strongly peaked and the electric field exhibits a typical saw-tooth pattern usually associated with non-linear wave steepening. Another important effect that can be observed in the non-linear case is a change in the wavelength of the plasma wave. As the amplitude of the wave increases so does its wavelength. In the limit γp 1 we get [16, 17]: λN p = λp

1. Emax /E0 1 (2/π)Emax /E0 , Emax /E0 1

(1.11)

where Emax is the peak electric field of the plasma wave and λp is the plasma collisionless skin depth, λp = 2π/kp = 2πc/ωp .

10


(a)

(b)

Figure 1.1: Plasma wave produced in the laser wakefield accelerator regime in 1D, with a laser pulse with duration L = λp = 45 µm for (a) a0 = 0.5 and (b) a0 = 2.0. Simulations were done using the OSIRIS code (see Chapter 5 for details).

1.3 Electron acceleration and Detuning

11

Another important parameter of the plasma wave is phase velocity, This is especially relevant for the determination of minimum injection energy (the energy threshold beyond which electrons get trapped by the plasma wave), maximum energy gain, and detuning length (the length it takes an electron to outrun the wave). In laser driven wakefields this velocity will be approximately the same as the laser group velocity in the plasma i.e. vp ' vg . In the linear regime the laser group velocity can be calculated from the 1D plasma dispersion relation ω 2 = c2 k 2 + ωp2 . A laser beam of frequency ω will have vg = c(1 − ωp2 /ω)1/2 , and γp = γg = (1 − vg2 /c2 )−1 = ω/ωp . For a laser pulse with a wavelength λ = 1.053µm propagating through a plasma with an electron density of n0 = 1019 cm−3 this gives γg = 10. Nonlinear corrections due to the quiver motion of the electrons have been calculated in 1D [18] in the limit ωp /ω 1 and require replacing ωp2 by ωp2 /γ⊥ where γ⊥ is the relativistic factor associated with the transverse (quiver) motion of the electrons, γ⊥ = (1 + a20 /2)1/2 . This results in an increase of the of the laser group velocity. The curvature of the laser wavefronts while it is being focused will cause the group velocity to be slowed down. This is an intrinsically 2D/3D effect and considering these corrections the group velocity is then given by [19]: γg ' (ωp2 /ω 2 + 2c2 /ω 2 w02 )−1/2

(1.12)

For a plasma density of n0 = 1019 cm−3 and a laser pulse with λ = 1.053µm focused down to a 10µm spot size diameter, this correction is negligible (γg ' 9.75) but for lower plasma densities, where the 1D limit gives much higher results (for n0 = 1016 cm−3 we get γg ' 330) this is an important limiting factor, bringing γg down by an order of magnitude (for the same density we get γg ' 44).

1.3

Electron acceleration and Detuning

The high amplitude plasma wave produced by the laser pulse can be used to accelerate electrons, provided these electrons have sufficient initial velocity to become trapped by this wave. The electron velocity vz will then increase and approach the speed of light i.e. vz → c. As we’ve just seen the plasma wave phase velocity will be vp < c so the electrons will eventually overtake the wave, and move into a region where the electric field is no longer an accelerating one. This will impose a limit on the energy the electron may gain and it is usually referred to as electron phase detuning. The detuning length, Ld is

12

Laser-Plasma Accelerator Concepts n0 [cm−3 ] E0 [MV/cm] EW B /E0 EW B [MV/cm] γg = k/kp Ld [cm] Wmax [GeV] γg3D Ld3D [cm] Wmax3D [GeV]

1016 96.2 25.1 2414 317.2 3360 8110 41.82 58.41 141.0

1017 304.29 14.1 4278 100.3 106.2 454.5 38.89 15.97 68.33

1018 962.25 7.84 7524 31.72 3.36 25.27 25.35 2.15 16.15

1019 5.5 × 1017 3043 714.6 4.25 9.14 12901 6507 10.03 42.77 0.106 8.24 1.37 53.59 9.76 30.04 0.100 4.06 1.29 26.42

Table 1.1: Laser plasma accelerator parameters for several plasma densities. Calculations were done using a λ = 1.053 µm laser focused to a 10 µm spot size diameter. The 3D values include the wavefront curvature correction.

defined has the length it takes an electron to go into an opposing phase to that of the plasma wave. A higly relativistic electron, ve ' c takes a time td = π[ωp (1 − vp /c)]−1 to do so. In this situation the detuning length will be [20]:

Ld = ctd ' γp2 λp

(1.13)

for γp 1. For a plasma wave of the form Ez = Emax sin[ωp (z/vp − t)] the maximum energy the electron can gain is then given approximately by [1, 21]:

Wmax ' eEmax Ld ' 2πγp2 me c2

Emax E0

(1.14)

Using the relativistic wavebraking limit (1.9) and the 3D group velocity (1.12), for a cold plasma with a density of n0 = 1019 cm−3 and a laser pulse with λ = 1.053 µm focused to a r0 = 10 µm spot size, we get a maximum energy gain of Wmax ' 1.3 GeV, on a detuning length Ld = 1 mm. The same energy gain would require roughly 13000 times the accelerating distance on a 100 MV/m RF LINAC. Table 1.1 shows these same calculations for a number of different densities. The rightmost column shows parameters relevant to resonant excitation of plasma waves using the L2I laser system.

1.4 Laser Propagation in Plasmas

1.4

13

Laser Propagation in Plasmas

The propagation of light waves is one the basic features present in Maxwell’s equations. The Helmholtz wave equation can be easily derived from Maxwell’s equations assuming a time dependence of the form e−iωt and taking the divergence of Ampere and Faraday’s equations. The solutions to Helmholtz equations in vacuum for plane waves are well know [22]: E(r, t) = EM ei(k·r−ωt) (1.15) B(r, t) = BM ei(k·r−ωt) where ω is the angular frequency of the wave, k is its wave vector and EM and BM are two complex constants related through BM = k × EM /c. The well known vacuum dispersion relation ω = kc, and E · k = 0 and B · k = 0, implies that this is a transverse wave. For a laser pulse EM is related to the normalized vector potential through: EM =

a0 cme ω e

2πa0 me c2 λe

=

(1.16)

EM [GV/cm] ' 32.1a0 λ−1 [µm] where λ is the laser wavelength. When considering a finite beam, we must satisfy yet another equation, the so called paraxial Helmholtz equation, that is derived by assuming our beam consists of a plane wave modulated through some complex amplitude, i.e., U(r) = A(r) exp(i(k · r − ωt)), and then solving for the envelope A(r). One important solution to this equation is called the Gaussian beam [23]. Assuming k = kb ez , i.e., the wave is propagating along the z axis we get: ρ2 ik w0 − w(z) 2 e e U(r) = A0 w(z)

ρ2 z+ 2R(z) −iξ(z) −iωt

e

(1.17)

where w(z) is the beam radius, w0 the beam radius in the focal plane, R(z) the beam curvature and ξ(z) is an additional phase retardation in comparison with a plane wave. These quantities have the following values: s w(z) = w0

1+

z zR

2 (1.18)

14


R(z) = z 1 +

z 2 R

z

ξ(z) = arctan r w0 =

z zR

(1.19)

(1.20)

λzR π

(1.21)

where zR is known as the Rayleigh range. The spot size diameter r0 is defined as 2w0 . The Rayleigh range defines√half the length of propagation where the beam radius w(z) is smaller than 2w0 and we usually define the depth of focus of a beam as the whole length where this happens, i.e., twice the Rayleigh range: 2zR =

2πw02 λ

(1.22)

In a plasma solving the Helmholtz equations becomes more difficult because we need to account for the induced currents and density fluctations. In the linear limit the solutions (1.15) are still valid provided we use the dispersion relation for electromagnetic waves propagating in a plasma: ω 2 = ωp2 + k 2 c2

(1.23)

Note that this means that only waves with ω > ωp can propagate in a plasma. For ω < ωp , k becomes imaginary and the wave will not propagate because the electron response time ωp− 1 shields out the field of such wave. For a light wave of a given wavelength λ we then define the critical density ncr as the maximum plasma density where an electromagnetic wave of a given frequency ω can propagate i.e. where ω = ωp : ncr =

πme c2 −2 λ e2 −3

(1.24) 21 −2

ncr [cm ] ' 1.1 × 10 λ [µm] The phase and group velocity for a light wave in a plasma will then be: ω vφ = = c k

ω2 ω 2 − ωp2

1/2 (1.25)


15

dω c2 vg = = =c dk vφ

ω 2 − ωp2 ω2

1/2 (1.26)

Note that vφ > c for all propagating solutions (vφ real), and therefore the group velocity never exceeds the speed of light. The relativistic wave factor γg is simply γg = (1 − vg2 /c2 )−1/2 = ω/ωp and the refractive index η = c/vφ = (1 − ωp2 /ω)1/2 .

1.4.1

Guiding

As we’ve just seen when a laser is propagating in vacuum it undergoes Rayleigh diffraction, i.e. the beam waist evolves according to (1.18). In this situation the beam will only interact with the plasma along a few zR . Comparing this value with the detuning length Ld we find that in general we have zR Ld . For example for a λ = 1.053µm laser focused to a r0 = 10µm spot we get zR ' 75µm. In a n0 = 5.5 × 1017 cm−3 plasma we have Ld ' 4cm or Ld /zR ' 530. This means that if we’re to take full advantage of the laser plasma accelerator some means of optical guiding needs to be used. The basic principle behind optical guiding in plasmas is the some one that is used on optical fibers. If the medium where the laser pulse is propagating has a refractive index η(ρ) that has a maximum on axis i.e. ∂η/∂ρ < 0. This will mean that the wave will propagate slower on axis, which will make the wavefronts to curve towards it, thus guiding the beam. For small amplitude waves propagating in a uniform plasma the refractive index in the linear limit was given above. However, if we consider a radially inhomogeneous plasma, or a large amplitude electron wave where the electrons reach relativistic velocities, the plasma frequency needs to be corrected, i.e., ωp2 → (ωp2 /γ)n/n0 , where γ is the relativistic factor associated with the electron motion. For ωp2 /ω 2 1 we can expand the refraction index to first order in ωp2 /ω 2 : η(ρ) ' 1 −

ωp2 n(ρ) 2ω 2 n0 γ(ρ)

(1.27)

This means that at least two categories of channeling mechanisms exist: one based on having a higher relativistic factor for the electrons on axis, and the other based on large spatial scale density variations with lower density on axis. The first one is commonly referred to as relativistic optical guiding [24, 25]. The leading order of electron motion is its quiver motion in the laser

16

Laser-Plasma Accelerator Concepts field i.e. p⊥ = mca. In this situation we have γ ' γ⊥ = (1 + a2 )1/2 . Neglecting other effects on the refractive index (i.e. n = n0 ) we then get: ωp2 η(ρ) ' 1 − 2 (1 + a2 )−1/2 2ω

(1.28)

Since the laser pulse will have a peak intensity on axis this results in a guiding refractive index profile. Calculations using the paraxial wave equation [24, 25, 26, 27] have shown that when the laser pulse exceeds a critical power, P ≥ Pc the relativist effects will prevent the laser pulse from diffracting. Analysing the paraxial wave equation a solution for the evolution of the laser spot size of a gaussian beam with a2 = a20 exp(−2ρ2 /w02 ) has been found [26] which predicts w2 /w02 = 1 + (1 − P/Pc )z 2 /zR2 , so the spot size will focus for P > Pc . Note that the spot will not focus indefinitely as it might seem from the previous result. The approximations that were done break up and in fact it can be shown that the spot size will eventually become constant or oscillate arround a constant value. For a circularly polarized gaussian pulse we have P/Pc = kp2 a20 w02 /16 or: Pc [GW ] ' 17ω 2 /ωp2

(1.29)

However, relativistic optical guiding is inefficient for short laser pulse where L . γp /γ⊥ . This is due to the fact that the plasma electrons will react on the inverse plasma frequency time scale, so relativistic guiding effects will only become important after the above length. Typically, relativistic optical guiding will only affect the body of long pulses, L > λp . The second group of channeling mechanisms relies on density variation of the background plasma density. One way of achieving this situation is through a preformed plasma channel with the appropriate density profile [28, 29, 30, 31, 32]. Consider a parabolic density channel of the form n = n0 + ∆n(ρ2 /ρ20 ) where ρ0 is the channel width. Neglecting other effects, the refraction index in this situation will then be: ωp2 η(ρ) ' 1 − 2 2ω

∆n ρ2 1+ n0 ρ20

(1.30)

Again, after analyzing the paraxial wave equation for a gaussian beam in the form a2 = a20 exp(−2ρ2 /w02 ) it has been found [33] that optimal guiding is achieved when w0 = ρ0 provided the channel depth is equal to the channel critical depth, ∆n = nc , where nc is given by:


17

nc = (πre w02 )−1 (1.31) nc [cm−3 ] = 1.1 × 1020 w0−2 [µm] where re is the classical electron radius, re = e2 /me c2 . If the channel is not matched (∆n 6= nc ) then the beam waist will oscillate about its matched value provided the channel is sufficiently broad. Simulations have shown [34] that the above channels effectively guide short (L ∼ λp ) high intensity (a0 ∼ 1.8) through several tenths of Rayleigh lengths, with minimal diffraction of the laser beam. Other channel profiles for optical guiding have been proposed like the so called leaky channel, where the channel density is parabolic up to a given radius and then drops off to zero linearly. This channel does not guide higher order transverse modes and can stabilize certain instabilities [35], such as small angle forward Raman scattering , self-modulation and and laserhosing. Another channel profile that has been proposed is the so called hollow channel profile [29, 30], where the density is zero for ρ < ρHC and n0 otherwise. Within this channel the transverse profile of the axial wakefield is uniform providing uniform acceleration of an injected beam. Another mechanism for optical guiding based on a density gradient is ponderomotive self-channeling. In this situation the radial ponderomotive force of a long laser plasma will expel electrons from the propagating axis thus creating a density channel, which will enhance the effects of relativistic self focusing. It can be shown, however, that for P < Pc , this effect alone will not guide the laser pulse. For lasers approaching the critical power P → Pc , guiding will be achieved predominantly by relativistic self-focusing. Detailed calculations [36, 37] have shown that ponderomotive self-channeling only has the effect of slightly lowering the critical power for self-focusing, which then becomes: P [GW ] ' 16.2ω 2 /ωp2

1.4.2

(1.32)

Laser-Plasma Instabilities

A laser beam propagating through a plasma can excite a number of laserplasma instabilities that disturb laser propagation and affect the performance of laser plasma accelerators. The main instabilities relevant to laser-plasma based accelerators are stimulated Raman scattering [3], self-modulation [38, 39, 40, 33] and laser-hose instabilities [41, 42].

18

Laser-Plasma Accelerator Concepts The stimulated Raman scattering (SRS) instability can be described by the resonant decay of incident photon into a scattered photon plus an electron plasma wave (plasmon) i.e. (ω0 , k0 ) → (ω, k) + (ωs ± ω, ks ± k). The two solutions are known as the Stokes wave (ω0 − ω, k0 − k) and the anti-Stokes wave (ω0 + ω, k0 − k). Linear analysis for this instability in the 1D limit can be found in [3] assuming no spatial evolution of the laser pulse, and a more complete analysis can be found in [39]. Of all the possible scattered waves two cases are of special relevance: backward Raman scattering and forward Raman scattering. In backward Raman scattering (BRS) the laser wave (ω0 , k0 ) decays into a plasma wave (ω, k) and a backward going scattered wave (ω0 − ωp , k0 − k) where ω ' ωp and k ' 2k0 . This is typically the fastest growing of the SRS instabilities, and is significant for laser-plasma accelerators for a number of reasons. At low laser intensities the scattered light can be used for determining the plasma density through the analysis of its wavelength (since ωs ' ω0 − ωp ). At high laser intensities however, it has been shown that the spectrum of the scattered wave broadens [43] so the plasma frequency can no longer be determined through this method. BRS can also be responsible for trapping background plasma electron as it grows on amplitude. The phase velocity of the BRS wave is vφ = ω/k = ωp /2k0 c so plasma thermal electrons can be trapped which results in heating the plasma and creating a fast electron tail on the temperature distribution. In forward Raman scattering (FRS) the scattered wave will propagate at a very low angle from the laser with an associated phase velocity vφ ' c [3]. This plasma wave can then be used to accelerate electrons. FRS can be the basis for the self modulated laser wakefield accelerator (described in detail in section 1.5.2) in 1D where a long, L > λp , laser pulse becomes modulated at λp and drives a high amplitude plasma wave. This instability can be understood by looking at a long uniform laser pulse propagating colinearly with an initially small amplitude plasma wave. The local group velocity of the laser (1.26) will be smaller in the regions where the plasma wave local amplitude δn is positive (δn > 0) and larger in the regions of opposite behavior. This will cause a modulation of the laser intensity that will be 90o out of phase with the plasma wave. The ponderomotive force resulting from this intensity modulation will then drive the plasma wave to larger amplitudes, resulting in the FRS instability. Another key instability affecting laser propagation in a plasma is the so called self-modulation instability [11, 12, 38, 39, 40, 33]. In this instability the envelope of a long laser pulse, L λp , with power P > Pc , becomes modulated at λp through the interaction of laser pulse with the wakefield it

1.5 Laser-Plasma Electron Accelerators created. Consider this laser pulse propagating in a plasma generating a small finite wakefield in the form δn = δn0 (ρ) cos(kp ζ) which modifies the plasma refractive index according to (1.27). In the regions where ∂δn/∂ρ > 0 we will get a focusing effect and in the regions where ∂δn/∂ρ < 0 diffraction will be enhanced. This creates an axially periodic density channel that will cause the laser beam to be modulated at λp which will then increase the amplitude of the plasma wave, resulting in the self-modulation instability. The instability may grow until the beam completely breaks up into a series of laser ”beamlets” of length ' λp /2, that will be guided through several Rayleigh lengths. The self-modulation instability is the basis for the self modulated laser wakefield accelerator in 2D and 3D, since the high amplitude plasma wave produced can trap and accelerate background or injected electrons to high energies. Finally on this group of laser-plasma instabilities we have the so called hose instability. This instability occurs when a small fluctuation of the centroid position with λ = λp of a long laser pulse is present. This fluctuation will drive an asymmetric plasma wave that will then enhance this centroid fluctuation. Simply speaking, the laser pulse body will try to align itself with the laser pulse head, overshoot, and oscillate with growing amplitude around the main optical axis. A more detailed physical picture and analysis of this instability can be found in [20] and [41, 42]. This instability will eventually destroy the laser pulse and limit laser plasma acceleration. It can be shown [20] that if the laser centroid is sufficiently smooth we can avoid significant levels of hosing.

1.5

Laser-Plasma Electron Accelerators

Several mechanisms have been proposed for the construction of laser-plasma accelerator. The most common ones are the laser wakefield accelerator (LWFA), the plasma beatwave accelerator (PBWA) and the self-modulated laser wakefield accelerator (SMLWFA), that differ among themselves in the way they drive the plasma wave. In the LWFA a single short (. 1 ps) high intensity (& 1018 W/cm2 laser pulse drives the plasma wave. The laser pulse duration is chosen so that L = cτL is approximately the plasma wavelength λp . Up until the early 90’s such lasers were not commonly available and the PBWA was though of as the best alternative. In this accelerator two long pulse laser beams of frequencies ω1 and ω2 are used to resonantly excite the plasma wave. The laser frequencies are chosen so that ω1 −ω2 ' ωp , so that the beating of the two laser pulses will interact resonantly with the plasma. Finally the SMLWFA uses a single single short (. 1 ps) high intensity (& 1018 W/cm2 laser pulse to excite the plasma wave. The laser pulse / plasma parameters are chosen

19

20

Laser-Plasma Accelerator Concepts so that the laser pulse duration is large in regard to the plasma wavelength, L > λp , and the power of this laser pulse exceeds the relativistic self focusing critical power, P > Pc . The laser pulse will undergo a self-modulation instability resulting in the pulse becoming modulated at the plasma period and generating a high amplitude plasma wave. The current state of the L2I laser system (τL ' 150 fs, P ' 5 TW, I ' 1019 W/cm2 and a0 ' 3.2) allows access to both the LWFA and the SMLWFA configurations. We present here a quick overview of these two laserplasma accelerators.

1.5.1

Laser Wakefield Accelerator

The laser wakefield accelerator (LWFA) was first proposed by Tajima and Dawson in 1979 [1]. In this laser plasma accelerator an intense laser pulse propagates in an underdense plasma ω ωp and expels electrons from the region of the laser pulse through the ponderomotive force, Fp ∼ ∇a2 . By matching the length scale of the laser pulse (and hence the ponderomotive force) to the plasma wavelength, L ∼ λp the excitation of large amplitude plasma waves can then be achieved. The phase velocity of these plasma waves is close to the group velocity of the driving laser beam [1, 44]. For an axially symmetric laser pulse the typical wakefield amplitude will be maximum when L ' λp /2. The wakefield amplitude will be proportional to the wavebreaking field, which means that operating at higher densities will produce higher accelerating gradients, although requiring shorter laser pulses. However, as shown above, for higher densities the laser group velocity is reduced an electron detuning occurs much sooner, which will limit the energy gain. Because the wakefield is driven by a single short (L . λp ) laser pulse to small variations on the laser pulse length and/or plasma density. Furthermore, the various instabilities affecting a laser pulse propagating in a plasma are greatly reduced which results in a very ”clean” wakefield. Figures 1.2 (a) and (b) show the wakefield and the plasma density for a LWFA simulation. Isosurface values for (a) are -0.9, -0.6, -0.3, 0.3 and 0.6 for blue, cyan, green, red and yellow, in normalized units. Isosurface values for (b) are 0.5, 1.2, 2.0 and 5.0 for blue, green, yellow and red, normalized to the background plasma density. This simulation was done using the OSIRIS framework in 3D for a 30 fs, 800 nm laser pulse. Both the plasma wave density and accelerating gradients show an axially symmetric profile, and we can see the plasma wave deteriorates after a few wavelengths. Electrons with sufficient thermal energy are trapped by the plasma wave generated by the laser pulse

1.5 Laser-Plasma Electron Accelerators

(a)

(b) Figure 1.2: Wakefield (a) and Plasma wave density (b) in the laser wakefield accelerator.

21

22


Figure 1.3: Momentum vs. Position in the laser wakefield accelerator.

and are continuously accelerated by gradients of the order of GeV/m. Figure 1.3 shows the phasespace (momentum vs. position) density for this simulation, clearly showing particle trapping and acceleration. Several experimental demonstrations of the LWFA have already been done. Experimentally the first evidence of a high amplitude plasma wave being produced by the LWFA technique is attributed to [45]. Other researchers [46, 47] have also diagnosed these high amplitude plasma waves using different optical interferometric techniques. Evidence of accelerated electrons has also been presented [48]. Several mechanisms can limit the energy gain on the LWFA accelerator. The key problem in the LWFA is laser diffraction: as shown above a laser propagating in vacuum will undergo Rayleigh diffraction, and it focal depth will only be twice the Rayleigh length. If some form of optical guiding is not introduced, the laser will remain at high intensities for a few Rayleigh lenghts and therefore limit the acceleration length to this short extension. Due to

1.5 Laser-Plasma Electron Accelerators

Figure 1.4: Laser field envelope in the self-modulated laser wakefield accelerator.

the ultrashort nature of the laser pulses in the LWFA self-channeling through relativistic self-focusing is not possible and a preformed plasma channel is usually necessary. Once the diffraction issue is overcome the next limiting factor will be electron detuning. This limiting factor cannot be overcome, but will still allow for energy gains of hundreds of GeV in lengths much short than those achievable in standard accelerator technology. For the LWFA pump depletion, where the laser pulse looses energy to the background plasma, and laser plasma instabilities are generally not important.

1.5.2

Self-Modulated Laser Wakefield Accelerator

The self-modulated laser wakefield accelerator (SMLWFA) [49, 50, 51] uses a single single short (. 1 ps) high intensity (& 1018 W/cm2 ) laser pulse to excite the plasma wave. The SMLFWA operates at higher densities that LWFA such that the laser power is greater the relativistic focusing critical power, P > Pc , and that the laser pulse is long in compared to the plasma wavelength, L > λp . In these conditions the laser will propagate through many Rayleigh lengths because of ponderomotive and relativistic self-focusing and undergo a self-modulation instability that will cause the laser envelope to become modulated at the plasma period and generate a high amplitude plasma wave. The SMLWFA technique presents some advantages over the standard LWFA. First of all the matching condition L ' λp is no longer necessary and operation is possible over a wide range of λ/lambdap . Second, there’s no need for a pre-formed plasma channel becomes the laser pulse will self-guide through relativistic self-focusing. And finally acceleration is usually be enhanced because

23

24


(a)

(b) Figure 1.5: Plasma wave density (a) and Momentum vs. Position (b) in the self-modulated laser wakefield accelerator after 2.1 ps of laser propagation.

1.5 Laser-Plasma Electron Accelerators of the higher densities used and the fact that through relativistic self-focusing the laser will reach higher intensities than through normal focusing. However, the higher densities also mean that the laser group velocity will be smaller so electron detuning will become an important limitation on the electron acceleration. Furthermore, the laser pulse will eventually diffract and limit the acceleration distance, which may not happen in a guided LWFA setup. Figure 1.4 shows the 2D envelope of a 1 ps, 1 TW, laser pulse in the SMLWFA. The simulation was done using the OSIRIS framework in 2D. The self-modulation instability causes the laser pulse to break into a series of laser ”beamlets” of length ' λp /2. The laser propagates through several Rayleigh lengths before diffracting. Figures 1.5 (a) and (b) show the plasma wave density and momentum vs. position phasespace density for the same simulation. Again, electrons with sufficient thermal energy are trapped by the plasma wave generated by the laser pulse and are continuously accelerated by gradients of the order of GeV/m, which can be clearly seen on figure 1.5 (b). Experimental evidence of the SMLWFA was first obtained by [50], and a large flux of accelerated electrons with energies above 44 MeV has also been observed [51]. Recently experimental measurements of energies above 200 MeV in this regime [52] have also been reported.

25

Chapter 2 Spectrometer Design One key element of the laser-plasma accelerator experiments that we propose to do is a diagnostic tool that allows us to characterize the energy distribution of the accelerated electrons. Several techniques (see for example [53] and references therein) have been developed for this purpose, but apart from some exotic suggestions most configurations rely on a classic magnetic spectrometer. In these systems, some kind of magnetic field distribution is used as a dispersive element that separates the electrons angularly according to their kinetic energy. The key reason for using an energy dispersive element instead of an energy resolving detector is that most energy resolving detectors will only work for a single incident particle, as we will see in the next chapter. Given the expected electron fluxes we therefore decided on using a magnetic dispersive system in connection with particle counting detectors. The design of such system must consider several aspects that are well know from the field of nuclear spectroscopy. The key characteristic for such system is that it must provide deflecting and focusing of the accelerated electrons so that particles with the same energy departing from the same (object) point will be focused onto a single (image) point. It must also allow for a broad range of energy to be analyzed simultaneously, so that the full energy spectrum of accelerated electrons in laser-plasma experiments can be characterized in a minimum number of laser shots. The system being designed in based on time proved deflecting magnet designs for broad-range energy spectrum analysis. In this chapter we present the design and construction of the dipole magnet used in the electron spectrometer built. We present a brief overview of the theory behind charged particle optics, and then proceed to the description of the design and construction of the system.

28

Spectrometer Design

2.1

Introduction

To determine the trajectories of charged particles within magnetic field it is necessary to know the spatial distribution of theses fields, calculate the Lorentz force acting on the particles, and solve the relativistic equations of motion. In the case of a dipole magnet the electric field is null; the equation of motion for a charged particle is then reduced to: p˙ = q(v × B)

(2.1)

where p is the linear momentum, q is the particle charge, v is the particle velocity and B is the local magnetic field. The second term on this equation is simply the Lorentz force acting on the particle. Since this force is perpendicular to the particle momentum, its module will not change. As a result, the relativistic mass of the particle will also be a constant of motion and we can rewrite (2.1) as: r˙ = v

v˙ =

q (v × B(r)) m

(2.2)

(2.3)

where r is the particle position and m is the particle relativistic mass. One situation of special interest is that of an uniform magnetic field e.g. B(r) = B0 eZ . In this case (2.3) have a well know analytical solution, where the charged particles have a circular motion on the x − y plane. The radius of curvature of their motion is the well know Larmor radius rL = p/(qB) that can be more conveniently expressed as a function of the relativistic kinetic energy T : p (T + m0 c2 )2 − m20 c4 rL = cqB

(2.4)

Where m0 is the rest mass and c is the speed of light. Again, as the relativistic mass of the particle is a constant no further changes are necessary.

2.1.1

Sector Field Lenses

The requirements for a magnetic dispersive system for the spectrometer we are building actually go beyond the simple separation of particles according

2.1 Introduction

Figure 2.1: Focusing of charged particles by a homogeneous magnetic sector field.

to their energy. The fact that we are not dealing with single particles but with particle beams, with finite widths and divergence, imply that some sort of focusing mechanism is required. Generally speaking, all particles emerging from a source point with the same energy should be focused onto a single image point. We are in fact looking for a form of magnetic lens, but where the chromatic (energy) aberration is not only desired, but also a key characteristic of the system. The simplest form of a magnetic system that achieves these requirements is a sector shaped magnetic field. A detailed analysis of the sector field lens, as it is generally known, will allow us to better understand the fundamental characteristics of these systems and provide us with the necessary know-how for the analysis of more sophisticated solutions to our problem. The behavior of sector field systems has been thoroughly explored, and the focal and dispersive properties of these systems are well known. Assume a magnetic sector field limited by abrupt edges such that the magnetic flux B0 is constant inside the sector and zero otherwise, as shown on figure 2.1. Although such a distribution violates Maxwell’s equations, and is thus unrealistic, the discontinuous change from B = 0 to B = B0 accurately describes most firstorder effects if an effective field boundary is defined as described in section 2.2.2. Let us assume that the charged particles enter and exit the magnetic field region perpendicularly to the sector edges. For convenience the x and y coordinates are defined in relation to the particle path, y being tangent to the particle trajectory outside the sector (and thus perpendicular to the sector edges) and x as being perpendicular to this trajectory. The optical axis of this system coincides with the y axis defined. The z axis is defined as being vertical, with its is origin placed half way between the two poles of the magnet generating this field. This sector has an angle of α, and the radius of curvature

29

30

Spectrometer Design of main the particle trajectory inside the sector is ρ, which obviously coincides with the Larmor radius, rL , of the particle. The distance from the source point A to the left edge of the sector is defined as l1 and the distance from the right edge of the sector to the focus point B is defined as l2 . A detailed analysis of this system can be found in [54]. By studying the plane of symmetry we find that a particle beam originating from A is deflected by an angle equal to that of the sector, α, and is focused on point B. According to Barber’s rule [55] these three points (A, B, M) will lie on the same line. In the chosen x − y coordinates the focusing behaviour of this system is similar to that of a thin lens placed at the center of the sector field with a focal length of: sin α 1 = fx ρ

(2.5)

Figure 2.2 depicts this analogy. Note that in the coordinate system used the beam deflection is not considered. The distances ’l10 e l20 are in general aproximated by l10 =l1 + δ and l20 =l2 + δ, where δ = ρtan(α/2) as seen on figure 2.1. All the usual relations from geometrical optics are also valid, like the relation between the image/object distances: 1 1 sin α 1 + 0 = = 0 l1 l2 fx ρ

(2.6)

and the transverse magnification, MT : xB l20 MT ≡ =− 0 xA l1

(2.7)

It should also be noted that, like in geometrical optics, this is a linear description and that we’re considering an unrealistic magnetic field. This aproximation is, however, perfectly adequate for the analysis required for the dimensioning of this system. Oblique entrance and exit angles In case the particle beam crosses the sector edge at an oblique angle, the individual particle trajectories will enter the magnetic field region sooner or later according to their coordinate x, as can be seen on figure 2.3, where 0 and 00 stand for the entry and exit angle, respectively. As a consequence all

2.1 Introduction

31

Figure 2.2: Equivalent system using a thin lens. Note that in this coordinate system beam deflexion is not considered.

Figure 2.3: The defocusing effect caused by oblique entrance and exit angles. A beam originating from A’ is focused in B’. For perpendicular entrance and exit angles the image-object relation would be between A and B. Note the definition of 0 and/or 00 .

trajectories entering the magnetic field with a negative x coordinate will be less deflected in the direction of the optical axis (more deflected for a positive x), which results in a defocusing effect. Similarly, should the angle 0 have a negative value, an increased focusing effect would be observed. A similar effect occurs at the exit boundary. Trajectories exiting the magnetic field with negative x coordinate will be less deflected in the direction of the optical axis (more deflected for a positive x). A particle with a trajectory parallel to the optical axis will be deflected by the first oblique region (defined by 0 ) by an angle ∆α. Considering x˙ = dx/dt, the particle transverse velocity, and v = dy/dt the particle longitudinal velocity, we get v sin(∆α) = x, ˙ or to first order: x˙ 1 ∆α ≈ = v v

Z

t1

t0

1 x¨dt ≈ ρ0

Z

0

dy = −x tan 0

x tan 0 ρ0

Remembering the ray optics equations for optical rays parallel to the optical axis entering a thin lens we see that the two defocusing effects can

32

Spectrometer Design therefore be described by adding a pair of thin lenses at the sector edges with focal lengths fx0 and fx00 given by: 1 tan ε0 = − fx0 ρ 1 tan ε00 = − fx00 ρ

(2.8) (2.9)

Equations (2.8) are also valid for negative 0 and/or 00 , which result in focusing effects. Focal line We now look at the ability of a sector field magnet to separate particles according to their kinetic energies. Ideally, such a system should focus all the particles onto a focal line, linearly separated according to their energy. Sector magnets are only seldom used in this type of problems because a wide range of particle kinetic energies will result in a very broad of trajectory radius, ρ. This large range of particle trajectories will cause a high variation of the exit angle 0 which means that, for each energy being analyzed, the dispersive element will have significantly different focal properties. Furthermore, only a reduced range of deflection angles can be used, since the particle trajectories intersect the focal line at a very low angle. For these reasons, sector magnets are only used as very short energy range discriminators, being set to a single energy by the geometry and magnetic flux intensity. However, these magnets can successfully be employed in integrating configurations, where an additional beam dump used together with the deflecting magnet serves as a high pass filter, selecting particles with kinetic energies above a certain threshold. The cutoff energy is defined by the system geometry and magnetic flux intensity and a full spectrum can then be obtained by sweeping the field intensity. For the design of such system, a detailed knowledge of particle deflection and focusing is then essencial. Consider the system in figure 2.4, for a sector angle of θ. A particle traversing the magnetic field region will undergo a deflection by an angle α, with a radius of curvature of ρ. Upon reaching the exit boundary it will have moved δX = ρ sin(α) in the horizontal direction, and δY = ρ(1 − cos(α)) in the vertical direction, where X and Y are cartesian laboratory coordinates. Solving for the radius of curvature as a function of the deflection angle, ρ(α), we get:

2.1 Introduction

33

Figure 2.4: Beam trajectories on a sector field for arbitrary deflection angles.

ρ(α) =

a0 sin α − (1 − cos α) tan θ

(2.10)

The effect of the oblique entrance and exit angles will be that of two thins lenses placed at the entry and exit edges of the sector. According to equation (2.8) we get for this geometry: 1 tan(−θ0 ) 1 tan(−θ00 − α) = − = − fx0 ρ(α) fx00 ρ(α)

(2.11)

By combining (2.10), (2.11), and the usual object-image relations from geometrical optics we get an analytical solution for the focal line of this system. We can see the results in figure 2.5. The full expression for the focal line is very complex and brings very little insight. A better analysis of this system can be obtained by assuming that the distances between the sector and the object and image are much larger than the distance between the sector edges, a0 . In this situation we can assume that the system is the association of three thin lenses in direct contact. The focal length of this triplet, fxT , will then be: 1 sin α − tan(−θ0 ) − tan(−θ00 − α) = fxT ρ(α)

(2.12)

In figure 2.6 we can see the focal length as function of the deflection angle for this system. Note that, just like a regular optical lens, this system

34

Spectrometer Design

Figure 2.5: Focal distance for a sector field as a function of the deflection angle, for the following parameters: a0 = 19.9 cm, θ0 = 10o and θ00 = 10o .

in unable to image an object placed closer than it’s focal length, which means that there is a minimum deflection angle, below which the system is unable to focus the beam.

Fringe-Field effects As stated above, the abrupt transition from a region with no magnetic field into a region with finite magnetic field flux clearly violates Maxwell’s equations. In a real dipole magnet, the field intensity starts to decrease before reaching the edge of the sector, and extends beyond the region defined by this edge. The exact behavior of the field depends on the construction details of the dipole magnet. The effect of this fringe field is that particles feel the effect of the magnetic field before reaching the actual sector region. It has been show that we can accurately model this effect to first order by placing a virtual boundary outside the sector, with edges parallel to the sector edges, and assume the field changes abruptly on those boundaries, as shown on figure 2.7. The actual position of these boundaries obviously depends on the actual fringe field behavior. Some empirical estimates show that placing them at a distance of 0.62D to 0.67D from the physical sector boundary, where D is the pole gap, gives good results. A detailed analysis of the fringing field is given on section 2.2.2. So far we’ve only concerned ourselves with particle trajectories in the x − y plane. In fact, neither in the field free region nor in the region of

2.1 Introduction

Figure 2.6: Focal length of a sector field as a function of the deflection angle for the following parameters: a0 = 19.9 cm, θ0 = 10o and θ00 = 10o .

Figure 2.7: Fringe Field. The real magnetic field from a dipole magnet is presented in comparison with the idealized magnetic field. The dashed line represents the virtual field boundary.

35

36

Spectrometer Design the homogeneous magnet does a particle experience forces in the z direction. However, as shown in [56], a particle crossing a fringing field at an angle , as shown on figure 2.7, experiences vertical forces, provided the particle velocity has a component along z. This happens because the curvature of the magnetic field implies that Bx and By are no longer null, and the resulting Lorentz force will then have a vertical component. To first order the deflections that then occur can also be described by thin lenses placed at the sector edges that exhibit a focusing behavior for positive angles. Astonishingly enough, these do not depend (to first order) on the detailed fringing field distribution and are simply: tan ε0 1 = fz0 ρ 1 tan ε0 = fz00 ρ

(2.13) (2.14)

We can use this effect for vertical focusing of the beam, reducing it’s height inside the gap of the magnet and increasing the particle flux in the detection region. We could even consider doing double focusing, i.e., focusing the beam simultaneously on the horizontal and vertical planes. However, It is not usual to achieve these results using a single focusing element although some geometries can achieve first order double focusing (see for example [57]), and the final result is usual a strongly stigmatic lens that only works for a short range of particle energies.

2.1.2

Particle Transport Systems

Another application of sector field lenses is that of transporting all electrons from the source point to a secondary point. This point is then used either either as the source point for a deflecting magnet like the ones described above, or for the placing of some energy resolving detector. As we will see in the next chapter, such detector systems rely on only one particle reaching the detector at a time, which renders them useless for the applications we’re interested in. However, we have considered using them to overcome the space difficulties that arise on laser-plasma accelerator experiments, allowing for the source point to be placed much closer to the actual spectrometer. Several geometries have been proposed for this purpose [58, 59, 60] and all rely on geometries close to a 270o dipole system, with or without some added entry angle. These systems are described as being triple-focusing, with

2.2 Broad Range Electron Spectrometers

Figure 2.8: Triple focusing dipole system.

stigmatic (focusing in both horizontal and vertical planes) and achromatic focusing (no energy dispersion). Figure 2.8 shows one of these systems. The detailed analysis of these system can be done with the equations described above and can be found on the given references. The main disadvantage of these systems resides on their size. Even for very large magnetic fields on the order of 2 T, the larmor radius for a 200 MeV electron is over 33 cm, which means a very large system would be required for the range of energies we’re interested in.

2.2

Broad Range Electron Spectrometers

Broad range magnetic spectrometers have been used successfully in nuclear spectroscopy for almost a century. Early systems consisted of a 180o sector magnet [61] and achieved high resolution and accuracy. These systems had

37

38

Spectrometer Design a few drawbacks: they could only record an energy range of about 1.1 to 1, they would only work with particles entering at a single angle and the required that both the source of the particles and the detector system be placed at the sectors edge. The first major improvement to broad range magnetic spectrometers came in the mid 1950’s with the development of circular shaped spectrometers, that came to be known as Browne & Buechner spectrometers after their 1956 paper [62]. These systems have been used successfully in nuclear reaction studies since their introduction and have already been used in the study laserplasma accelerator studies, and have the advantage of requiring a relatively small region of finite magnetic field in comparison with other geometries. One problem with the Browne & Buechner is that it provides no vertical focusing, resulting in low signal to noise ratios. One solution to this problem was proposed by Elbek and co-workers in the early 1960’s [63]. It’s similar to a dipole magnet with roughly 200o , but has a special entry angle to take advantage from fringing field effects. This system has very low aberration in all points of its extended focal line, and provides a net gain in intensity from the focusing in the fringing field. Finally, in the late 1960’s, the split-pole design was introduced [64]. It’s a two dipole system, the first with two curved faces and the second with a standard sector shaped magnet, and it provides near perfect focusing on both planes in the focal line. The design of this system only became possible through the use of computer simulation, which represented a major step forward in these designs. After careful consideration, we chose the Browne & Bruechner spectrometer model over the other systems mentioned mostly because of it’s smaller size and the fact that the system is much easier to manufacture. These two factors also resulted in a more economical solution, while maintaining the wide energy range and necessary precision that we stipulated as the basis for this project.

2.2.1

Circular Shaped Magnetic Fields

Circular shaped magnetic fields were first proposed in the mid 1950’s as a solution for a broad range magnetic spectrograph for nuclear reaction studies. In these systems a wide range of deflection angles is available while maintaining the same exit angle and focal properties. The analysis of this system can be done using the basic equations described on the above sections. Further analysis will then be made through numerical simulation, and detailed comparison between these results is presented.


39

Figure 2.9: Circular shaped magnetic field. For each deflection angle α, this system behaves as a sector field with the same angle.

Focal Line The equation defining the focal line for a circular shaped magnetic field can be easily obtained from those defining the focal line for a sector magnet. As shown on figure 2.9 we see that for each deflection angle α, this system behaves as a sector field with the same angle. Defining ρ0 as the radius of the magnetic field region, the deflection angle will be: α = 2 arctan

ρ0 ρ

(2.15)

Note that we are ignoring the curvature of the entry and exit faces which is consistent with the first order approach used to determine the general equations of sector optics. In fact it can be shown that this curvature corrects some second aberration effects and is indeed desirable. Using (2.5) we get, after some algebra: 1 1 − cos α = fx ρ0

(2.16)

Using geometrical optics we then get the relation between image and object distance: q e = ρ0 1 − e cos α

(2.17)

40

Spectrometer Design where e is defined as: e=

p p − ρ0

(2.18)

This expression defines the focal line for our system. Equation (2.16) is a polar equation defining a second degree curve; for e > 1 we get an hyperbole. This hyperbole intersects the boundaries of the magnetic field for q = ρ0 or for a deflection angle of:

αM

ρ0 = arccos − p

(2.19)

The angle αM defines the maximum deflection angle for which particles are focused outside the magnet, and therefore defines the valid range for equation (2.16). For deflection angles above αM the system will no longer behave as a circular field system, and the focal properties are lost. One interesting characteristic of this system is that all beams, after being deflected, appear as if they are coming from a virtual source placed at the center of the magnetic field. This is extremely useful in the alignment of the detection system as it greatly simplifies the determination of particle trajectories after leaving the magnetic field region. Positive Entry Angle A small variation on this system consists on adding a small angle at the entry edge of the magnetic field [65], as presented on figure 2.10. The equations that we just derived are still valid provided we include the effect of a thin defocussing lens. As shown in figure 2.11 this can be done by changing the object distance to the system. Again using geometrical optics we find that: l0 =

l 1 − l/fd

(2.20)

where fd is the focal length of the defocussing lens we just added, and is therefore defined negative. As expected we get l0 < l. Now using (2.8) we get: tan ε0 tan (α/2) l l =l 1+ ρ0 0

−1 (2.21)


41

Figure 2.10: Circular shaped magnetic field with an entry angle. This angle provides some vertical focusing.

Figure 2.11: Defocussing lens effect. The object appears to be closer than it really is because of the increase in beam divergence.

In this system the lens-object distance is given by l = p − ρ0 so finally we get for the apparent distance to the center of the magnet p0 : p 0 = ρ0 + 1+

p − ρ0 tan 0 tan(α/2)(p−ρ0 ) ρ0

(2.22)

If we now combine this with (2.17) we can determine the analytical expression for the focal line. After some algebra we get: q=

ρ0 1 − cos(α) − 1 +

p−ρ0 ρ0 +tan 0 tan(α/2)(p−ρ0 )

−1

(2.23)

In this situation no approximations can be made because the distances

42

Spectrometer Design involved are of the same order as the field size. Should we decide to include fringe-field effects, we need only to define a virtual boundary for our field as described above. To do so it is sufficient to replace ρ0 in the above equation by ρ∗0 = ρ0 +∆D, where again D is the gap distance and ∆ is a value between 0.62 and 0.68. The focus line defined by (2.23) can be seen together with results from numerical simulations in figure 2.17.

2.2.2

Numerical Simulation for Circular Fields

Another method of analyzing these systems is through the use of numerical simulations. Using a field model as accurate as possible, we solve the relativistic equations of motion for the particles in three dimensions. This will allow not only for the validation of the assumptions made in the preceding sections, but also for the appropriate inclusion of fringing field effects. A new code was written specifically for this purpose, based on a 4th order Runge-Kutta algorithm. In the simulations described below, the time step used was of 0.1 ps, which corresponds to a particle motion of about 30 µm for the energies of interest. Fringe Field Expression The accurate modeling of this system requires a detailed knowledge of the magnetic field distribution. The key element for this to be achieved is getting a detailed description of the fringing magnetic field near the edges of the magnet pole pieces. A semi-empirical formula for this field can be found, for example, in [66]. Again using D for the air gap, x0 as the distance from the physical pole edge, and defining s0 as s0 = x0 /D we have: h(s) =

1 Bz,0 = B0 1 + exp(S)

(2.24)

where B0 is the constant induction well inside the pole piece region and Bz,0 is the induction in the fringing field, as measured in the median plane (z = 0). The parameter S is given by the power series: S = c0 + c1 s + c2 s 2 + c3 s 3

(2.25)

where s is again the distance along the x0 direction in units of D, but the origin for s has been placed at the virtual field boundary i.e. s = s0 − s0 . Two

2.2 Broad Range Electron Spectrometers Field Type Short Tail Long Tail

s0 c0 0.62 0.3835 0.68 0.531

43

c1 c2 c3 2.388 -0.8171 0.200 2.341 -0.7799 0.110

Table 2.1: Definitions for Short-Tail e Long-Tail fringing fields.

Figure 2.12: Fringe field calculated from (2.24) for the two situations presented. Note that the horizontal axis is in units of D, the air gap distance.

situations are of special interest: the short tail curve and long tail curve (Table 2.1). These two situations model the limiting values for standard magnet assemblies, and can be see on figure 2.12, plotted from (2.24). Figure 2.13 shows the magnetic field used for the simulations of the magnet, for a gap size of 13 mm. In the median plane (z = 0) these expressions completely describe the magnetic field. Outside this plane however, the magnetic field will have x and y components. For a simpler analysis we can describe these field components as a Taylor series along z. For symmetry reasons the expression for Bz can only depend on odd powers of z, and the expressions for Bx and By can only depend on even powers of z. To first order we then have: Bz = Bz,0

(2.26)

We can now use the following Maxwell’s equation, ∇ × B = 0, to determine the remaining values for the field. Using (2.26) and the coordinates

44

Spectrometer Design

Figure 2.13: Complete magnetic field for the modeling of the magnet. The entry angle is 10o and the field radius is 11.5 cm. The blue solid lines represent the physical boundaries of the poles pieces.

defined on figure 2.14 we get (again, to first order): ∇×B=0⇒

z Bx = Bx,0 + 1!

∂Bx ∂z

=z

z=0

∂Bx ∂Bz = ∂z ∂x

∂Bz,0 ∂x

=−

dh z B0 sin (β) D ds

(2.27)

And for the field component along y: By =

z dh B0 cos (β) D ds

(2.28)

As defined earlier, y is tangent to the motion of the beam, so no Lorentz force arises from this field component. The x component however is perpendicular to the motion of the beam, and is therefore responsible for a vertical force that acts on the beam. This force is responsible for the vertical focusing effect described earlier. Figure 2.15 shows the horizontal components of the fringing field in the same coordinate system from figure 2.13, for a vertical position slightly above the median plane.


45

Figure 2.14: Coordinate system for the calculations of the fringing field horizontal components.

Figure 2.15: Horizontal Fringe Field. (a) and (b) show the X and Y components of the fringing field for a vertical coordinate of 0.02 D normalized to the magnetic flux in the uniform field region. The dimensions for the magnet are the same as in figure 2.13.

Angular Dispersion Using the distribution for the fringing field determined in the previous section we can now proceed to the numerical determination of the focal and deflecting properties of this system. The key characteristic of this system is it’s ability to disperse angularly particles having different energies. A seen on (2.4) particles with different kinetic energies will have different trajectory radius. Combining this with (2.10) we can get the full expression for the deflection angle as a function of kinetic energy, magnetic flux intensity and magnetic field radius: 



ρ0 Bcq  α = 2 arctan  q 2 2 4 2 (T + m0 c ) − m0 c

(2.29)

46

Spectrometer Design

Figure 2.16: Deflection angle for electrons for magnetic fields from 0.5 T to 3 T. The magnet has a radius of 11.5 cm, and an air gap of 1.3 cm. Calculations were done using the long-tail model. The solid line shows the analytical solutions, and the symbols show the numerical solutions.

In the strongly relativistic limit, T m0 c2 , this expression reduces to:

B α = 2 arctan cqρ0 T

(2.30)

This means that for high enough energies the dispersive properties of this system depend directly on the fraction (ρ0 B)/T . Simply put, a particle with twice the energy will have the same deflection if we either double the magnetic field intensity or double the magnet size. On figure 2.16 we show the angular dispersion for electrons on this geometry. The differences between numerical and analytical solutions are negligible which means that the analytical expression is perfectly adequate for dimensioning purposes. Note that we did not present results for a fringing field of the long-tail type because the difference from the short-tail results was less than 0.5%, and was not visible on the scale used on the plot. However, these differences are significant, and when using this system it is recommended that results from simulations done with the measured distribution of the magnetic field are used.


Figure 2.17: Focal lines for a circular magnet with an entrance angle. The magnet has a radius of 11.5 cm, an entry angle of 10o and an air gap of 1.3 cm. The particles originate from a point 85 cm away from the magnets entry angle. The solid line shows the analytical solutions, and the symbols show the numerical solutions. Calculations were made for three fringe field models: ideal, short-tail and long-tail. On this scale the differences between short and long tail are not visible.

Focal Lines

We now analyze the focal lines of our system. Figure 2.17 shows the numerical and analytical solutions for the focal lines, for several fringing field models. For determining the focal points numerically we used the intersection of two particles that originated with a ±0.5o angle from the optical axis. We took the final positions of the particles, about 1 m from the exit boundary of the magnet, and determined the intersection of the two straight lines originating from those positions that were tangent to the particle velocities. It is clear from the results that the effect of the fringing field is simply to move the focal line slightly away from the one calculated with an abrupt magnetic field edge, which is consistent with the virtual fielf boundary approximation described above. Note that the results obtained inside the magnet are invalid for both the numerical and analytical solutions. In both these cases we’re assuming that the particles follow straight lines after passing through the focal point.

47

48

Spectrometer Design Lateral Magnification Should the particle source not be a single point, as it is expected, it is important to look into the lateral magnification properties of our system. Just like an ordinary lens imaging system, the dipole magnet will image the particle source on an image focal line. This image focal line contains the previously discussed focal point for the energy being analyzed and is perpendicular to the optical axis. The expression for the lateral magnification of single thin lens imaging system is well known [22]. The total magnification for a combination of multiple lenses can be easily deduced by remembering that the total magnification of the system will be the product of each individual magnification. For a system of two thin lenses, with focal lengths of f1 e f2 and separated by a distance d we get: MT =

f1 si2 d (so1 − f1 ) − f1 so1

(2.31)

where so1 and si2 are the distances between the object and the first lens and the image and the second lens. For our system with a circular magnetic field with an entrance angle we then get: MT =

fε0 q 0 ρ0 (p − ρ0 − fε0 ) − fε0 (p − ρ0 )

(2.32)

This obviously depends on the deflection angle. The effect of the entrance angle is again considered by using the corrected distance q 0 . We can also determine the relation between the particle beam divergence and the lateral magnification. Defining βo as the initial (object) beam divergence and βi as the final (image) beam divergence we get: tan(βi /2) = |MT | tan(βo /2)

(2.33)

We use the modulus of the magnification because, like with any focusing lens, the magnification is negative i.e. the image appears inverted. Figure 2.18 shows the lateral magnification for our system, determined both numerically and analytically. For the numerical calculations we used (2.33) for determining the magnification. Once again the differences between analytical and numerical results are negligible. Also note that the magnification is always smaller than unity. Since the object size (accelerating plasma) will always be . 100µm, the final beam size will be very small and we will have very high energy selection precision on the detectors.


Figure 2.18: Lateral Magnification as a function of the deflection angle. The magnet has a radius of 11.5 cm, an entry angle of 10o and an air gap of 1.3 cm. The particles originate from a point 85 cm away from the magnets entry angle. The solid line shows the analytical results and circles show the numerical results.

Aberration All analytical calculations done above are only a first order approach to the focusing properties of these systems. Higher order effects cause imaging aberrations, where particles with the same energy originating with different angles in regard to the optical axes, or with different distances from the optical axis, are focused on different positions. The deviation from the first-order calculations is small but must nevertheless be considered in determining the resolution and precision of our system. A detailed analysis of the aberration of magnetic lens systems can be found in [54] and references therein. The analytical study of these aberrations is very complex and goes beyond the scope of this work. A numerical study is, however, straightforward. We analyze the variation of the focusing point distance p for a particle going parallel to the optical axis with the distance from the optical axis before entering the magnetic lens. We then define the longitudinal aberration of the system G as the ratio between the variation in the focal point position p − f0 and the focal length of the system f0 for a given deflection angle:

49

50

Spectrometer Design

Figure 2.19: Longitudinal aberration for a system with a 11.5 cm radius, an entry angle of 10o and an air gap of 1.3 cm.

p − f0 G= f0 α

(2.34)

Figure 2.19 shows the results. To maintain an aberration of less than ±2.5% throughout the whole deflection range we must limit the entry opening size to less than 4 cm. If we only use deflection angles from 50o to 90o opening sizes of up to 10 cm are acceptable. Also note that the aberration is not symmetric, and has opposite sign depending on the distance to the optical axis.

2.3

Dimensioning

Having studied the fundamental equations governing these systems we now proceed to the dimensioning of the deflecting magnet. The key parameter in dimensioning a spectrometer of this type is the range of energies we propose to analyze. Given the current state of laser-plasma accelerators, we decided upon an energy range going from 10 to 200 MeV. Also, as stated above, we want to be able to measure as much of this spectrum as possible simultaneously, without

2.3 Dimensioning

Figure 2.20: Deflection angle for 200 MeV electrons as function of the magnet radius ρ0 for several values of the magnetic field. The source point is placed 1 m away from the entrance of the magnet.

resorting to accumulating multiple laser shots. Another key parameter to take into account is the particle flux at the detectors. To maximize signal to noise ratio a double focusing system would be ideal, but some degree of vertical focusing should be enough. According to (2.29) the deflection properties of this system depend solely on the construction parameters B (the magnetic field flux) and ρ0 (the radius of the magnetic field). Magnetic flux densities of above 2.4 T (Iron magnetic saturation field) are difficult to achieve and require the use of exotic metallic alloys, so we can use this value as an upper limit for the magnetic flux intensity and proceed from there. Figure 2.20 shows deflection angles for 200 MeV electrons as function of the magnet pole piece radius ρ0 for magnetic fields in the 1 T range. In deciding these two parameters we must also take into account the distance from the focal point to the magnet q. Increasing values of ρ0 will mean both high values and variation for q at low deflection angles (< 60o ), which are obviously not practical. Figure 2.21 shows this behavior for several values of ρ0 . Although larger values of q mean a longer focal line, and thus increased energy resolution, values above 1.2 m are not practical. For integration in the L2I laboratory the magnet entrance must be placed more than 1 m away from the center of the vacuum chamber where the interaction takes place, so

51

52

Spectrometer Design

Figure 2.21: Focal point distance to the center of the magnet (q) as a function of the deflection angle for several values of the magnetic field radius ρ0 .

our system must also take this into account. The horizontal collection angle required is 2o full angle, which corresponds to 3.5 cm opening at a 1 m distance. Regarding vertical focusing we decided on a 10o pole face entry angle, which is a typical value for systems designed for these energies. This angle is a balance between the required vertical focusing and the corresponding horizontal defocussing, which causes the focal line to be placed further away from the magnet. For low energy particles this is an added bonus because it allows for larger deflection angles to be used, but for high energy particles this causes the focal point distance to be even larger. The value chosen results in a good balance between these effects. It does not provide vertical focusing (as the distance between the source point and the magnet is smaller than the vertical focal length) but provides some degree of beam collimation. The vertical collection angle required is 0.25o full angle or 0.87 cm at a 1m distance. Assuming the particle would then travel 30 cm inside the magnetic field region this would mean that the air gap should be at least 1.13 cm, but because of vertical focusing a air gap of 10 mm is sufficient. After considering several parameter combinations and costs, we decided upon the values presented on table 2.2. Taking the fringing field effects into consideration these values result on a 200 MeV electron being deflected 43o and a 60 MeV electron being deflected 105o , using a 2.2 T field value. The energy range of the spectrometer is then 3.33:1.

2.3 Dimensioning Magnetic Flux Intensity (B0 ) Homogeneous Field Radius (ρ0 ) Air Gap (D) Pole Piece Entry Angle (0 ) Deflection Angles to Use (α)

53 2.3 T (Max) 11.5 cm 1 cm 10o 120o − 40o

Table 2.2: Magnet Parameters chosen.

Figure 2.22: Final pole piece design for our electron spectrometer.

We also chose to fit our system with a secondary sector magnet configuration. This will allow for the use of our magnet as standard sector field integrating configuration, which will be accessible by simply reverting the magnetic field. For reasons of symmetry we chose a 20o sector angle, given the 10o entry angle we had already stipulated. Figure 2.22 shows the final pole piece design. This 20o sector magnet has central trajectory radius of about 57.5 cm. Given the high intensity magnetic flux and uniformity requirements, magnet design is greatly simplified by having a return yoke that surrounds the pole pieces as much as possible on every direction. For use in the integrating spectrometer configuration we chose to embed an exit window into the magnetic yoke that is 4 cm wide, and allows for very high energy particles to exit from the system.

Chapter 3 Detection System An electron spectrometer system, such as the one we propose to build, consists mainly of two elements: a dispersive element and a detection system. The previous chapter presented the dispersive system in detail; we now proceed to describing the detection system. The general operation of the detection system required is that of an electron counter that must operate in the full energy spectrum that is available from the dispersing magnet. Multiple types of detectors have been used for this purpose. For the range of energies we are interested in most detection systems are either based on scintillation detectors or on solid-state detectors. These detection systems convert the energy deposited by the charged particle to either photons or electron-hole pairs that then require some form of signal processing to convert them into calibrated electron counts. These electron counts must then be collated and cross-referenced with the dispersing magnet parameters, thus obtaining the full electron energy spectrum. In this chapter we begin by presenting a brief overview of the theory behind charged particle detection, and then proceed to the description of the design and construction of our detection system.

3.1

Charged Particles Detection

The use of any radiation detector will basically depend on the way the radiation to be detected interacts with the materials in the detector. The understanding of the response signal from a detector must then be based on the knowledge of the fundamental mechanisms by which radiation interacts with matter. A

56

Detection System standard reference covering this field is the text by Evans [67]; we will only focus on fast electrons.

3.1.1

Coulomb Collisions

The fundamental mechanism for the interaction of charged particle radiation with matter is that of coulomb forces between their charge and that of the orbital electrons surrounding the absorbing matter atoms. While interactions of the particle with the nuclei of the material being traversed are possible, they are very rare, and have very little influence in detector response. Upon entering an absorbing medium, the charged particle radiation will immediately begin to interact simultaneously with multiple electrons. Depending on the interaction (collision), the impulse the electron of the medium receives may be enough either to raise the electron to a higher lying shell within the absorber atom (excitation) or to completely remove the electron from the atom (ionization). The energy being transferred to the absorber electron comes necessarily from the incident charged particle which will therefore decelerate as it crosses the absorbing medium. These collisions can also result in the deflection of the trajectory of the incident charged particle, but this effect is generally negligible for high energy electrons. For the analysis of the interaction of charged particles with matter it is usual define the linear stopping power S simply as the loss of energy, dW , of the charged particle per unit length, dx, of absorbing material, i.e.: S=−

dW dx

(3.1)

The calculations for the stopping power due to exciting and ionizing collisions were first done by Bohr using classical arguments [68] and later by Bethe, Block and others using quantum mechanics. The Bethe-Block formula for the stopping power due to collisions for a charged particle as a function of the momentum transfer is [69]:

−

dW dx

= 2π c

Z Na re2 me c2 ρ

1 A β2

2me γ 2 v 2 Wmax 2 ln − 2β I2

(3.2)

where Na is Avogadro’s number, Na = 6.022 × 1023 mol−1 , re is the classical electron radius, re = 2.817 × 10−13 cm, me is the electron rest mass, c is the speed of light, ρ is the density of the absorbing material, Z is the atomic

3.1 Charged Particles Detection

57

number, A is the atomic weight, z is the charge p of the incident particle in units of the electron charge e, β = v/c, γ = 1/ 1 − β 2 . Wmax is the maximum energy that can be transferred on a collision, which corresponds to an impact parameter of zero. For a charged particle of mass M me we get: Wmax =

2me c2 β 2 γ 2 q me 1 + 2 M 1 + β 2γ 2 +

m2e M2

(3.3)

where I stands for the mean excitation potential, and is the main parameter for the Bethe-Block formula. The calculation of this parameter is very difficult because it relies on unknown parameters for most materials. Based on several experimental measurements of dE/dx a semi-empirical formulas for I were determined. One such formula is [69]: I Z I Z

= 12 + Z7 eV Z < 13 = 9.76 + 58.8 Z −1.19 eV Z ≥ 13

(3.4)

which gives an idea of the evolution of I with the atomic number. This formula is however not accurate enough and improved values for I from experimental values are generally used. As we can see there is a linear dependence of the stopping power with the atomic number Z so in orther to minimize collisional losses it is usual to resort to low Z materials. Two corrections are generally done to the Bethe-Block formula: the shell and density corrections, in order to best fit experimental observations. The corrected formula becomes: −

dW dx

= 2π c

z2 A β2

Z Na re2 me c2 ρ

2me γ 2 v 2 Wmax C 2 ln − 2β − δ − 2 I2 Z (3.5)

These two corrections are especially important for low and high energies. The density effect comes from the fact that the electric field of the charged particle tends to polarize the atoms along its path, which results on shielding effects, that prevents electrons far from the path from feeling this electric field. Collisions with these electrons will therefore contribute less to the total predicted energy loss. The values for δ, the density effect correction parameter, are [70]:   0 4.6052 X + Cδ + a (X1 − X)m δ=  4.6052 X + Cδ

X < X0 X0 < X < X1 X > X1

(3.6)

58

Detection System where X = log10 (βγ), and X0 , X1 , Cδ , a e m depend on the absorbing material. Cδ depends on the mean excitation potential I and the absorbing medium plasma frequency, and, together with the remaining constants, are defined empirically. The shell correction accounts for effects which arise when the velocity of the incident particle is comparable or smaller than the velocity of the orbital electrons. At these energies we cannot assume the orbital electrons are stationary with regard to the incident charged particles, and therefore the Bethe-Bloch formula is no longer valid. This correction is generally small, and can be done through the following empirical formula, which is valid for βγ ≥ 0.1:

C (I, η) = (0.422377 η −2 + 0.0304043 η −4 − 0.00038106 η −6 ) × 10−6 I 2 + + (3.850190 η −2 − 0.1667989 η −4 + 0.00157955 η −6 ) × 10−9 I 3 (3.7) where C(I, η) is the shell density correction parameter, η = βγ and I is again the medium excitation potential in eV.

3.1.2

Energy Loss for Electrons

For electrons, due to their small mass, an additional mechanism for energy loss becomes important: the emission of electromagnetic radiation resulting from scattering in the electric field of the nucleus of the asborbing material, generally known as bremstrahlung. From a classical point of view this radiation can be understood as resulting from the fast electron acceleration when it collides with the absorber nucleus. At low energies of one MeV or less this process has very little influence. However, as the energy of the fast electron increases, the bremsstrahlung grows rapidly, and at energies of a few tens of MeV this energy loss mechanism begins to dominate. The total energy loss for fast electrons is then given by the sum of collisional and radiative losses: dW = dx

dW dx

+

c

dW dx

(3.8) r

The collisional energy loss term is similar to the Bethe-Block formula described above. However some small changes need to be done, because i) the low mass of the electrons and ii) the collisions are now between identical particles which are indistinguishable. This results in changing several terms in


59

equation (3.2). In particular, the maximum energy transfer on each collision is now Wmax = T /2, where T is the kinetic energy of the incident electron. The Bethe-Block formula for fast electrons then becomes:

−

dW dx

=

Z 2πNa re2 me c2 ρ

1 A β2

c

2 τ (τ + 2) C ln + F (τ ) − δ − 2 Z 2 (I/me c2 )2 (3.9)

where τ is the kinetic energy of the incident electron normalized to the rest mass energy of the electron, me c2 , and F (τ ) is defined as: 2

F (τ ) = 1 − β +

τ2 8

− (2τ + 1) ln 2 (τ + 1)2

(3.10)

and the remaining quantities are the same as defined on equation (3.5).

3.1.3

Bremsstrahlung

The radiation energy loss, bremsstrahlung, depends on the strength of the electric field felt by the fast electron because of the positive nuclei of the absorbing material. We must then take into account the shield effect from the orbital electrons. This shielding effect can be parameterized by the quantity: ξ=

100me c2 hν Wi Wf Z 1/3

(3.11)

where Wi is the total energy of the incident election (kinetic + rest mass), Wf is the total energy of the incident electron after the collision, and hν the energy of the emitted photon, Wi − Wf . This parameter is related to the ThomasFermi atom model and is small, ξ ≈ 0, for complete shielding and large, ξ 1 for no shielding. The energy loss through radiative effects is calculated by integrating the bremsstrahlung cross section times the photon energy over the allowable energy range, i.e.: −

dW dx

Z =N

r

ν0

hν 0

dσb (E0 , ν) dν dν

(3.12)

where N is the numerical atom density of the absorbing material, N = ρNa /A, and ν0 = Wi /h. We can rewrite (3.12) as:

60

Detection System

−

dW dx

= N Wi Φrad r

where Φrad

1 = Wi

ν0

Z

hν 0

dσb (Wi , ν) dν dν

(3.13)

The motivation for this is related to the fact that the bremsstrahlung cross-section, dσb /dν is approximately proportional to 1/ν. The integral Φrad is therefore practically independent of ν and depends only on the absorbing material. The exact expressions for dσb /dν are extensive and its derivation goes beyond the scope of this work. We can, however, get the expressions for Φrad on two important limits: ξ 1, no screening, and ξ ≈ 0, full screening. For low energies, me c2 Wi 137me c2 Z 1/3 , ξ 1, we have no screening and the integral Φrad gives [71]: Φrad = 4Z (Z +

1) re2 αs

1 2 E0 − ln me c2 3

(3.14)

where αs is the fine structure constant, αs = 1/137. For high energies, Wi 137me c2 Z 1/3 , ξ ≈ 0, we have complete screening and the integral Φrad now gives [71]:

Φrad = 4 Z (Z +

1) re2 α

1 −1/3 ln 183 Z + 18

(3.15)

It is interesting to compare equation (3.15) with the ionization energy loss formula (3.9). Whereas the ionization loss varies logarithmically with energy and linearly with Z, the radiation loss increases almost linearly with energy and quadratically with Z. Figure 3.1 shows the stopping power for fast electrons for helium and aluminum, for comparison of the Z dependence of both energy loss mechanisms. Note that the stopping power is normalized to the density ρ, so that the two curves have roughly the same scaling. For low energies the collisional mechanisms dominate while for high energies the radiative processes are dominant Material data and constants were taken from the NIST ESTAR database [72, 73]. Table 3.1 shows density, mean excitation potential and atomic number, Z (effective Z for compound elements), for common materials.


Figure 3.1: Total, collisional and radiative stopping power for fast electrons for Helium and Aluminum. The dotted red and blue lines show the collisional and radiative energy loss. The solid black line shows the total stopping power. Note that the stopping power is normalized to the the density, ρ.

Material ρ [g/cm3] I [eV] Z Air, Dry Near Sea Level 1.205 × 10−3 85.7 7.37 Aluminum 2.6989 166.0 13 Carbon (Graphite) 1.7 78.0 6 CsI 4.51 553.1 54.02 Copper 8.96 322.0 29 Helium 0.166 × 10−3 41.8 2 Iron 7.874 286.0 26 Lead 11.35 823.0 82 Phosphorus 2.2 173.0 15 Mylar (Polyethylene Terephtalate) 1.4 78.7 6.46 Polystyrene 1.06 68.7 5.61 Teflon (Polytetrafluorethylene) 2.2 99.1 8.28 Silicon 2.33 173.0 14 Table 3.1: Density (ρ), Mean Excitation Potential (I), and Effective Atomic Number (Z) for several absorbing materials.

61

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Detection System

Figure 3.2: Total stopping power for Mylar, Air and Helium.

For direct comparison figure 3.2 and table 3.2 show the total stopping power for a number of materials. Note that the stopping power for Helium is about on order of magnitude lower than for Air. This is especially useful as a low cost alternative to vacuum chambers when transporting electrons through air is not feasible due to scattering and energy loss. Another important quantity is the so called CSDA range, R. This quantity is a very close approximation to the average path length traveled by a charged particle as it slows down to rest, calculated in the continuousslowing-down approximation. In this approximation, the rate of energy loss at every point along the track is assumed to be equal to the same as the total stopping power. The CSDA range is obtained by integrating the reciprocal of the total stopping power with respect to energy, i.e.:

ZW0 R=

dW dx

−1 dW

(3.16)

0

Where W0 is the rest mass energy of the electron, W0 = me c2 . Figure 3.3 and table 3.3 show this parameter for a number of materials. We see that for the energy range we’re interested (10 - 200 MeV) the CSDA range is very large in air, several tens of meters, and again much larger for Helium.


Material Air Aluminum Carbon CsI Copper Helium Iron Lead Phosphorus Mylar Polystyrene Teflon Silicon

0.1 0.00438 8.59600 7.31400 10.21064 24.32640 0.00067 22.20468 22.79080 6.99820 5.37040 4.28028 7.53720 7.62842

dW/dx [MeV/cm] Energy [MeV] 1 10 100 1000 0.00202 0.00260 0.00581 0.03510 4.01057 5.18459 15.00858 114.54132 3.23800 3.77200 7.96800 49.48000 5.35788 10.00318 56.42010 538.49400 11.72864 17.92896 78.31040 700.04480 0.00030 0.00036 0.00059 0.00219 10.62990 15.78737 65.26759 575.11696 12.74605 27.31945 179.10300 1764.92500 3.26700 4.34940 13.41560 105.66600 2.40940 2.79020 5.92900 36.97400 1.91224 2.18572 4.36720 25.82160 3.42540 4.06560 9.65140 65.45000 3.56723 4.69029 14.01961 108.78770

Table 3.2: Total stopping power for several absorbing materials, for electron energies from 0.1 MeV to 1 GeV.

Figure 3.3: CSDA Range, R, for Mylar, Air and Helium.

63

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Detection System

Material 1 Air 407.70 Aluminum 0.21 Carbon 0.25 CsI 0.16 71.06 × 10−3 Copper Helium 2.73 × 103 Iron 78.22 × 10−3 69.1 × 10−3 Lead Phosphorus 0.25 0.34 Mylar Polystyrene 0.42 0.24 Teflon 0.23 Silicon

R [cm] Energy [MeV] 10 100 1000 3 3 4.31 × 10 26.28 × 10 81.58 × 103 2.17 11.88 30.31 2.84 18.76 58.45 1.37 4.75 8.98 0.69 2.89 6.07 30.13 × 103 0.22 × 106 0.95 × 106 0.77 3.36 7.21 0.54 1.67 2.97 2.63 13.78 33.99 3.84 25.28 78.50 4.86 33.19 107.92 2.67 16.51 47.45 2.42 12.96 32.48

Table 3.3: CSDA range, R, in centimeters, for several absorbing materials, for electron energies from 1 MeV to 1 GeV.

3.2

Scintillators

Ionizing radiation detection through the use of scintillating light is one of the oldest techniques on record. The scintillation process remains however as one of the most useful methods available for the detection and measurement of ionizing radiation. Scintillating materials are generally divided into to two groups: organic scintillators and inorganic scintillators. Table 3.4 shows several properties of some scintillating materials used for β − spectroscopy. The scintillating process in organic materials is related to the transitions in the energy level structure of a single scintillator molecule and can be described independently of the physical state of the scintillator (solid, liquid or gas). Energy absorption by the material can result in the excitation of both vibrational and electronic levels. From these excited states the electrons will then decay onto excited vibrational states of the ground state, which results in the emission of the principal scintillation light, that generally lies in the visible range. Because these transitions have a smaller energy than transitions going to the absolute ground state, these materials are generally transparent to their own scintillating light. Since the scintillation process for organic scintillators is independent of the physical state of the scintillator, organic scintillators can be packaged in a

3.2 Scintillators

Material Type NaI(Tl) Crystal CsI(Tl) Crystal CsI(Na) Crystal NE102A Plastic NE110 Plastic NE216 Liquid NE250 Liquid BCF-20 Fiber

65 Density λmax [g/cm3 ] [nm] 3.67 415 4.51 540 4.51 420 1.03 423 1.03 434 0.89 425 1.04 425 1.05 492

Light Yield [Photons/MeV] 38000 52000 39000 10000 9200 12000 7700 8000

Table 3.4: Physical properties of common Scintillators.

number of different ways. One of the most commonly used organic scintillator is obtained by dissolving an organic scintillator in an appropriate solvent, and is generally available packaged in a glass container. As for solid scintillators, another common application relies on dissolving the scintillator in another solvent that can then be polymerized. These plastic scintillators are very easy to handle and inexpensive. These materials are generally used for the spectroscopy of low energy electrons because, having both low densities and atomic numbers, they reduce backscattering of incident electrons. For example, a typical plastic scintillator will only backscatter about 8% of incident electrons [74]. They also present very high efficiency in the sense that almost all of electrons entering the scintillator will result in the production of detectable light. For inorganic materials, the scintillation mechanism depends on the energy levels defined by the crystal lattice of the scintillator. As usual, energy absorption may result in the elevation of an electron from its normal position in the valence band, leaving a hole in the normally filled valence band. In a pure crystal, the return of the electron to the valence band through a photon emission is a very inefficient process and furthermore, the energy gap is usually too large for the frequency of the emitted photon to lie in the visible wavelengths. To enhance the probability of visible photons during the deexcitation process it is usual to add small quantities of impurities generally called as activators. These impurities create special sites in the crystal lattice where the normal energy band structure of the material is altered, creating energy states inside the forbidden energy gap through which the electron can deexcite back to the valence band. Since the energy of this transition is smaller than the scintilator crystal band gap it can now originate visible photons and serve as a basis of the scintillation process. A charged particle passing through the

66

Detection System detection medium will create a high number of these electron-hole pairs as a result of its collisional energy loss inside the material. The positive holes will quickly drift to the location of the activator atom and ionize it because the ionization energy of the impurity is less than that of the scintillating crystal. Meanwhile, the electron is free to move through the crystal lattice until it finds an ionized activator. At this point the electron can drop into the free activator energy level, creating a neutral activator configuration which can have its own set of excited states. If the activator state formed is an excited state with an allowed transition to the ground state then its deexcitation will quickly occur with a high probability of photon emission. A careful choice of the activator material will result on visible transition energy. At very high electron energies the use of inorganic scintillators becomes advantageous [69] because the energy loss is mainly through the production of bremsstrahlung and the subsequent electron showers that it produces. In this situation a high-Z material is needed in order to improve radiation loss, and inorganic scintillators, with much higher density and atomic number become preferable.

Light Output Only a small fraction of the collisional energy loss of the charged particle is converted into light. The remainder of this energy is dissipated non-radiatively, mainly in the form of lattice vibrations or heat. The scintillation efficiency is defined as the fraction of the energy loss that is converted into light, and it depends not only on the type of particle but also on its energy. In some cases the scintillation efficiency is independent of the particle energy, which results in a linear dependence of the light yield on initial energy. Scintillator response to charged particles can be described by the relation between dL/dx, the scintillating light yield per unit length, and (dW/dx) the energy loss for the particle. In the absence of second order effects that would reduce scintillating efficiency we can assume a linear relation, i.e.: dW dL = S(W ) dx dx

(3.17)

Assuming that S(W ) will not vary significantly while the electron loses energy inside the scintillating material (which is generally the case for high energies), the total amount of scintillating light emitted per unit of absorbed energy is a constant depending only the incident energy, i.e.:

3.3 Semiconductor Diode Detectors

dL = S(W ) dW

3.3

67

(3.18)

Semiconductor Diode Detectors

Semiconductor detectors are based on crystalline semiconductor materials, usually silicon and/or germanium. The first prototypes of this type of detectors began to be used in the early 1960’s, providing high resolution detectors for energy measurement, and are considered to be the future of detection systems for high energy experiments. The basic operating principle of semiconductor detectors is similar to that of scintillating detectors. The passage of ionizing radiation will create electron-hole pairs (instead of electron-ion pairs) which are collected by an electric field and converted into an electric current. The advantage of semiconductor detectors is that the required energy for the creation of an electron-hole pair is usually much smaller that the energy required to excite a scintillator. The amount of pair creation will be around an order of magnitude higher than on scintillating detectors. Furthermore, these detectors have a high density, which means that they especially fit for the detection of high energy electrons, and the output signal of the detector is directly measurable by an electronics system, not requiring some form of light conversion. However, with the exception of Silicon detectors, the small band gap generally requires some form of cooling to make the detector usable, and the higher sensitivity makes them susceptible to electronic noise. The most commonly used silicon detectors for charged particle measurement are the surface barrier detectors (SBD). These detectors rely on the junction formed between a semiconductor and certain metals, usually n-type silicon with gold or p-type silicon with aluminum. Because of the different Fermi levels on these materials (half way energy between the valence and conduction band) a contact electric field arises when when the two materials are put in contact. This will cause the lowering of the band levels in the semiconductor, which results in a np type junction behavior. All electron-hole pairs produced in the depletion region will be accelerated by the depletion zone electric field and produce a signal. Electron-hole pairs being generated elsewhere will recombine roughly in the same position where they are formed and therefore will not contribute to the detection operation. SBD’s can be made with varying thickness and depletion zone regions. Furthermore, an external bias can be applied to the detector, varying the depletion zone thickness, increasing the versatility of these detectors.

68

Detection System

330 o K 77 o K

Si 3.62 eV 3.81 eV

Ge 2.96 eV

Table 3.5: Average energy required for electron-hole pair creation in silicon and germanium.

The detector characteristics of semiconductor detectors are very similar to those of scintillating detectors. The average energy needed to create an electron-hole pair at a given temperature is found to be independent of the type and energy of the radiation and only depend on the type of semiconductor material used. Table 3.5 shows the average energy necessary for the creation of an electron-hole pair in silicon and germanium at normal and liquid nitrogen temperatures. It is also worth mentioning that the intrinsic detection efficiency of semiconductors is close to 100% as very few particles will fail to create some ionization in the sensitive volume. We can the establish a direct relationship between the deposited energy and the deposited charge: e dW dQ = dx Wavg dx

(3.19)

where Wavg is the average energy required for the creation of an electron-hole pair. Assuming that dW/dx will not vary significantly while the electron loses energy inside the semiconductor detector, the total deposited charge reduces to: e dQ = dW Wavg

(3.20)

By coupling a charge sensitive amplifier (i.e. an integrating amplifier) to the SBD, and knowing the length of the depletion region we then have a direct measurement of the electron flux.

3.4

Electron Detectors Used

The detection system for the electron spectrometer being built needs to provide both high resolution and precision in measuring electron flux and the ability to analyze the full range of energies available from the electron magnetic spectrometer. This detection system is meant to work as an electron counter only; all energy separation is done by the magnetic field described in the previous

3.4 Electron Detectors Used chapter. After careful consideration of the price and performance of detection systems available and the expected electron fluxes we chose to use two complementary detection systems in our spectrometer, one to provide continuous monitoring along the full detection range of the magnetic spectrometer and another to provide high sensitivity and precision for a set of well defined energies within the same range. Both these systems are based on scintillating materials, mainly because these provide a lower cost solution, while maintaining the required performance. The first of these systems is a mechanically flexible scintillating plate that will be placed along the focal plane of our spectrometer, and the second is an array of scintillating detectors placed immediately after the scintillating plate at well defined angles.

3.4.1

Scintillating Plate

The scintillating plate being used must fulfill two basic requirements: first of all it must cover the whole focal plane of the magnetic spectrometer, preferably placed exactly along it, and second, it must allow the electrons to cross it with a minimal energy loss, so they can later be detected by the second detection system described in the next section. This means that we require some kind of thin, long scintillating plate, preferably based on a flexible material, that can be easily matched to standard imaging equipment, so that sensitivity and signal to noise ratio are enhanced. The usual answer to these issues is to use some type of phosphor deposited on a thin transparent support, usually glass or some form of plastic. Several companies supply these materials (see [75] for example) but these are usually custom items and are relatively expensive. Recently it has been suggested [65] that using intensifying screens from X-Ray imaging products provides an efficient and relatively inexpensive solution. These screens are built for use in medical X-Ray imaging applications as high efficiency intensifying screens that convert X-Ray photons into visible light, and work like any other scintillating detector. They allow for the use of standard panchromatic film for X-Ray imaging instead of standard X-Ray film, which results in both lower cost and lower exposures to radiation for the subject being imaged. We chose the CURIX ORTHO REGULAR screens from Agfa [76], which can be purchased individually as replacement units for damaged X-Ray cassetes, and comply with the requirements stated above. The specific type we chose has peak emission at 540 nm, which is matched to the peak efficiency of standard CCD cameras and panchromatic film, which we plan to use for imaging. Figure 3.4 shows the scintillating plates prior to mounting in the appropriate stands for use with the electron spectrometer system. These screens

69

70

Detection System

Figure 3.4: CURIX ORTHO REGULAR screens from Agfa used for electron detection and counting.

can be cut to measure and can cover the whole spectrum using just two stripes of scintillating material.

3.4.2

CsI(Tl) Scintillators

For the high-precision, high-sensitivity detectors the two normal options are SBD detectors or scintillating crystals coupled to a light collection device. SBD devices present the advantage of simple use and calibration and also high sensitivity. However, they are a much more expensive solution, and have an higher sensitivity to noise than the latter solution. Furthermore, the associated electronics is also complex and expensive, having to deal with high biasing voltages with ultra-low leakage currents, and also easily subject to saturation. The use of scintillating crystals was usually discarded because they required bulky and fragile photomultiplier tubes for normal operation. However, recent advances in the the development of semiconductor photodiodes allow us to build compact, high efficiency detectors, consisting of scintillating crystal attached to a photodiode. Although being less sensitive that SBD’s (for the same detector thickness) these detectors offer a lower price and a greater simplicity in signal processing. The photodiodes can generally operate without biasing, which reduces leakage current, and their quantum efficiency is extremely high,

3.4 Electron Detectors Used

Figure 3.5: CsI crystals mounted on photodiodes. The left detector is presented without the protective wrapping for clarity.

assuring that 60%–80% of scintillating light will produce an electric signal. For these reasons we decided to build an array of scintillating crystals coupled to PIN photodiodes for the second detection system. We chose CsI(Tl) as the scintillating material because of its high density and scintillating efficiency, and also because its scintillating light spectrum is maximum in the red region of the spectum (λmax = 540 nm). These crystals were grown to our specifications and have 5 × 5 × 30 mm in dimension. This cross section was chosen as a balance between total collected flux and energy resolution and the length was chosen for stopping most electrons with energies . 100 MeV. The photodiode chosen was the Hamamatsu model S1227-66BR PIN photodiode [77], whose quantum efficiency is maximum in the same region of the spectrum where the scintillating crystal has maximum emission. These photodiodes have very low capacity and therefore better signal to noise ratio. Photodiode dimensions were chosen to be slightly larger than the crystal cross section in order to facilitate mounting the crystals on the photodiodes. Figure 3.5 shows the detectors built, with and whitout the protective wrapping. Some precaution is necessary while handling CsI(Tl) crystals because these are very toxic and somewhat hygroscopic, which means that in contact with skin moisture they can dissolve and penetrate the skin through the pores. For better light collection the CsI(Tl) crystals were first polished on the side and top edges to maximize internal reflection, and ground finely on the bottom edge to minimize reflection on the transmission to the photodiode.

71

72

Detection System The CsI(Tl) crystals were then glued to using special optical glue with a peak transparency in the appropriate region of the visible spectra, and an optical index halfway between CsI and the photodiode window. This glue needs to be cured using UV light so special positioning of the crystal in relation to the photodiode had to be maintained while gluing so that the UV light could penetrate into the glue. We also took care in having the glue overflow over the crystal walls for better light collection. We then proceeded to wrapping the CsI(Tl) crystals in a layer of fiber glass which refractive index enhances light reflection into the crytal, followed by a teflon tape to hold everyhting in place and allow for safe handling of the detectors. Teflon was chosen because it is a low density plastic and is commonly available in very thin tapes, thus minimizing the stopping power of the wrapping material. The output of the detector is then analyzed by a charge-sensitive amplifier / peak detector pair that is described in detail in the next chapter. Other detection systems were also considered, and special interest was given to scintillating fibers [75]. These optical fibers are built with a plastic scintillator core, and the sinctillating light produced in them is guided by the fiber, which provides a simple way of transporting the scintillating light to a light collection device. They present the advantage of being relatively inexpensive and also being easily coupled with a CCD camera for data acquisition. Furthermore, for low energies, they present two additional advantages: they are built of a low density plastic scintillator which, as we’ve seen above, is preferable for low energy electrons, and they can be used in vacuum, which again is critical for low energy operation.

Chapter 4 Electron Spectrometer System We present in this chapter the integration and assembly of our electron spectrometer system. The large size and weight of the magnet presented some mechanical difficulties that required building a custom transport cart and positioning stand. We also had to consider the use of a vacuum box for the spectrometer system for operation at low energies. Finally we discuss the integration of the detection system, with special attention to the signal processing and acquisition electronics developed for this purpose, as well as the computer interface.

4.1

Deflecting element

The deflecting magnet described on chapter 2 was custom built by Bruker [78] according to our design. We chose a custom magnet with non interchangeable pole pieces instead of a standard model with customizable pole-pieces because for the very high magnetic flux densities we require this meant a drop on the magnet cost by a factor of 4. The magnet is built from two symmetric solid blocks of iron that form both the pole-pieces and the magnetic return yoke. These blocks are later assembled together with the magnet coils. The manufacturing process starts with a solid block of iron. Using a milling machine, iron is removed to define the pole piece shape and to add room for the coils. A 20 mm chamfer was added to the pole pieces for added magnetic field uniformity and the pole piece faces were then trimmed to a mirror like surface. This construction technique has the added benefit of allowing for the addition of a vacuum box for using this system with low energy electrons by separating the two halves of the magnet, placing the vacuum box between them and rejoining

74

Electron Spectrometer System the magnet. The total weight of the magnet (iron core and pole pieces plus coils) is about 670 kg, and all the handling of the magnet is done using the 2T crane at the L2I laboratory, that attaches to 3 points on the top half of the magnet where special attachment loop holes can be inserted. Figure 4.1 shows the deflection magnet placed in front of the main L2I vacuum chamber and a detail of the pole piece. The coils are made of 5 turns of copper each. The relatively small number of turns is justified by the high operating current this magnet uses, going up to 500 A. They are built with high purity copper for low resistance and heat dissipation, presenting a 27.7 mΩ nominal resistance. However, at these high currents, a strong power dissipation is inevitable, so these coils have a special cross section allowing for internal and external cooling using water. An array of thermal sensors attached to the coils acts as an interlock switch that turns off the power supply if overheating occurs. For full power operation the cooling system requires a minimum flux of 25 l/min of water. Special attention was also given to the connection cables between the power supply and the coils, especially in terms of magnetic shielding. The power supply used is a BLMPS MON 1 16/500 C4 unit also from Bruker, shown in figure 4.2. It is a current stabilized unit, supplying a maximum of 500 A and 16 V using a nominal power of 8 KW, with a stability ∆I/I better than 10−5 . This unit requires an additional cooling flux of 10 l/min of water.

4.1.1

Magnet Stand

Given the large weight of our system a custom stand for the magnet for had to be designed and built. This platform needed not only to be able to properly support the near 700 kg of the magnet, but also to properly align it with the experimental setup at the L2I laboratory. This meant placing the symmetry plane between the two pole pieces at a height of 1.2 m, with a precision of . 0.5 mm and making sure the magnetic field is perpendicular to the optical axis of the main L2I beam (which will also be the main axis of propagation for accelerated electrons) again with high precision. Figure 4.3 shows the 3D rendering CAD model of the spectrometer stand designed to fulfill these needs that includes a mock model of the magnet, as well as the constructed magnet stand. This platform was built mainly in iron and the elevating of the magnet to the appropriate height is achieved through the use of a tubular structure with a 50 mm thickness. The tubular structure was chosen because it has increased torsion strength and lower weight and vibration. This structure is welded to a triangular base where the height and tilting positioning threads are also attached, and we’ve added three small bulkheads for better vibration

4.1 Deflecting element

75

(a)

(b) Figure 4.1: The deflection magnet for the electron spectrometer system placed in front of the main L2I vacuum chamber (a) and a detail of the pole piece (b). Note that for (b) we separated the two magnet halves.

76

Electron Spectrometer System

Figure 4.2: The BLMPS MON 1 16/500 C4 power supply (foreground) for the deflecting magnet (background).

control. The vertical tilting and alignment is achieved through three bronze M45 bolts with a square head. Bronze was chosen for these bolts because of its greater hardness that results in less friction and easier handling. These bolts end with a reversed cone shape that stands on stainless steel spheres than then stand on stainless steel plates. Each of these three plates has a different design for added stability and ease of alignment using the standard dot, dash, plane design type. On the top of this structure a 30 mm iron plate was welded to serve as the actual magnet and instrumentation stand. A thin plate of bronze is placed on top of the iron plate again to minimize friction with the magnet when aligning the system horizontally. Finally, four horizontal positioning controls for horizontal translation and rotation aligment, made out of stainless


Figure 4.3: 3D rendering of the magnet stand, showing the alingment and positioning elements (a) and the actual stand with the deflecting magnet on top (b).

steel and using M8 bolts for positioning, are placed on top of the magnet stand.

4.1.2

Operation

The deflecting magnet has a peak magnetic field of 2.23 T when used with an electrical current of 500 A. Figure 4.4 shows the measured values of the magnetic field as a function of the electrical current. Up to 100 A (' 1.6T), the field grows linearly with the electrical current but for higher values magnetic saturation begins to occur, resultoing in a slower growth of the magnetic field. The magnetic field flux intensity follows approximately the following expression:

77

78


Figure 4.4: Magnetic field as a function of supply current.

B(I) =

17.32I + 0.350 I ≤ 50A mT 4434 exp − 21.15 − 2036 mT I > 50A I

(4.1)

which differs from the experimental measurements by less than 1%. The solid red line on figure 4.4 shows the results from (4.1). The magnet was also tested for field homogeneity and distribution. Inside the 20 mm chamfer region the field homogeneity ∆B/B is better than 0.5 % for field values up to 1.6 T, and 1.8 % at the maximum field 2.2 T. This degradation in uniformity is expected and is related the the magnetic saturation of iron. Figures 4.5 and 4.6 show the field distribution near the magnet edges, for the entrance and exit faces, respectively. For the exit face two deflection angles, 58o and 120o , (figure 4.6 (a) and (b), respectively) were chosen and measurements were taken along the main optical axis of those trajectories. Also note that the coordinate X represents the distance to the mechanical field boundary (i.e. the iron edge), so that for negative values we are inside the mechanical field boundary region. All measurements were taken for three values of the magnetic field representing 100%, 50% and 30% of the maximum field attainable (2.23 T, 1.11 T and 0.67 T, respectively). For the two lower values of the magnetic field the behavior of the field distribution is very similar for both cases. Because we don’t have any magnetic field shunts, the magnetic field extends for a larger region than the one predicted by (2.24) and (2.25). In our situation the field is best described


79

Figure 4.5: Magnetic flux density near the entry edge of the pole piece for three field intensities.

s0 B-Field sR Low Intensity -1.92669 Saturation 0.70825

c0 d0 -12.61503 0.86387

c1 c2 d1 d2 11.16563 0.18061 0.76723 -0.13073

c3 – 0.01083

Table 4.1: Fitting parameters for the measured magnetic field.

by replacing (2.25) by:

S=

d0 + d1 s d 2 s>1 d0 + d1 + d1 d2 (s − 1) s ≤ 1

(4.2)

defining s = s0 − sR and using the values on table 4.1. Figure 4.7 shows the fitting function against measurements for the 30% field intensity. For the highest value of the magnetic field, the field intensity actually falls faster than predicted by (4.2), again due to the magnetic saturation of the iron. In this situation, (2.25) becomes valid again provided we use the fitting parameters shown on table 4.1. Figure 4.7 also shows the results for the 100% field intensity.

80


(a)

(b) Figure 4.6: Magnetic flux density near the exit edge of the pole piece for three field intensities for 58o (a) and 120o (b) deflection angles.


81

Figure 4.7: Fitting curves for the distribution of the magnetic field.

Effective field boundary and deflection angle The deflection angle as a function of particle energy and magnetic field can now be determined through numerical simulation using the measured values for the magnetic field. A set of simulations was done covering all operational values, with energies ranging from 10 MeV to 200 MeV and field intensities of 0.1 T to 2.3 T. Figure 4.8 shows the actual field used in the simulation along with a few trajectories and focal points. Results for the deflection p angle are shown in figure 4.9 as a function of the parameter χ = B/p = B/ (T + m0 c2 )2 − m20 c4 , i. e., the ratio between the magnetic field and electron momentum. The shaded areas represent trajectories that exit the magnet outside the operating parameters. For accurate treatment of experimental data, the deflection angle as a function of χ should be obtained by interpolating these results. Another approach is to determine effective field boundary (EFB) for the magnet and use the theoretical formulas developed in chapter 2. The EFB can be found by fitting the theoretical expression for the angular deflection (2.29) to the simulation simulation results. Rewriting (2.29) as a function of χ we get: α = 2 arctan (ρ0 c e χ)

(4.3)

Results are shown in figure 4.10 where we only considered the focal points that lie outside the mechanical field boundary. The numerical fit is

82


Figure 4.8: Simulation field from measured values, sample trajectories and focal points.

ρ0 11.9262

EF B 0.4262

ρ0s 11.8316

EF Bs 0.3316

Table 4.2: Effective field boundary. Values are in cm.

very good and only deviates slightly from the simulation results for the higher magnetic field to momentum ratios. For simplicity we can also approximate χ by χs = B/T which is only valid for T me c2 . This is very useful as a rule of thumb for using the spectrometer, since it allows for quickly setting new magnetic field intensities while looking for a specific energy. Results are shown in figure 4.11. As expected the simulated values no longer stay along a smooth line, reflecting the problems at low energies. However, the results are still quite good, and for simple calculations this method represents a valid ”quick and dirty” alternative. Table 4.2 summarizes the results obtained.


Figure 4.9: Deflection angle as a function of χ = B/p.

Figure 4.10: Effective Field Boundary Calculation. Simulation results shown as black circles and numerical fit as a red line. Calculations were done for B = 50%.

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84


Figure 4.11: Effective Field Boundary Calculation for the simplified parameter χs . Simulation results shown as black circles and numerical fit as a red line.

Focal Line For the highest precision in energy discrimination, detectors should be placed along the focal line for our system. Figures 4.12 and 4.13 show the results for a point source placed at 1.2 m from the center of the magnetic field. These results match closely the EFB calculations using the values determined in the previous section (figure 4.12). The expression for the EFP calculations for the focal point distance is lenghty, so interpolating from these simulation results is advised. Depth of focus Another issue that is relevant to the placement of the detection system is the so called depth of focus of our system that we define as the length of beam path where the spot size is below a given dimension. This term is used here in a slightly different meaning than in standard imaging optics, where the image position remains constant and we refer to the object distances where it remains in focus. Calculations were done by determining the linear lateral magnification and then determining the spot size along the exit path based on a given entry slit width.


Figure 4.12: Focus point distance to the center of the magnet. Simulation results shown as black circles and the EFB calculations as a red line.

Figure 4.13: Depth of focus for the magnetic system. The particle beam as a spot size below S × W between the two corresponding lines.

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86

Electron Spectrometer System Figure 4.13 shows simulation results for the depth of focus. The simulation parameters were the same as the ones used in the previous section. Results are shown as a function of the ratio S between the entry slit width W , and the horizontal spot size. Note that the results inside the pole piece region are not valid, as they assume the beam is no longer being deflected. As a we can see using an entry slit of a given width L will result in the beam being focused to a spot size smaller than this width L for a very long path. This is to be expected, as at this distance the magnetic system will behave closely to a collimating lens. As a result, when it is unpractical to place the detection system along the focal line, we can use a narrow entry slit and still maintain a high level of precision. A good compromise is also achieved by using a slit width over spot size ratio of 0.5 which greatly reduces the distance from the region where the beam is focused to the center of the magnet, especially for high energies.

4.2

Vacuum box

As discussed on the previous chapter a fast electron will interact with matter reducing its energy and deflecting its path. For high enough energies, the energy loss and deflection are negligible but for the low energies a careful study is required to determine if the use of a vacuum box is needed. The physical models presented in the previous chapter, although providing good qualitative and quantitative insight into the energy loss mechanism, are unable to account for particle deflection. To accurately model the latter numerical simulations are required. Simulations were done using the GEANT4 code [79]. This is a Monte Carlo code for the simulation of physics detectors and physical processes, and was developed as a toolkit for the simulation of the passage of particles through matter. Its application areas include high energy physics and nuclear experiments, medical, accelerator and space physics studies. In its previous versions it has been successfully applied to the development of high energy physics instrumentation for over two decades. Figure 4.14 shows the simulation results for an electron beam traveling through 70 cm of vacuum, crossing a 0.1 mm thick mylar window, and then traveling through 1.3 m of air. These parameters were chosen as they represent what would be typical operating parameters for an experimental spectrometer setup operating without a vacuum box. Mylar was chosen for the window of the main vacuum chamber as it presents low density and high mechanical

4.2 Vacuum box

Figure 4.14: GEANT4 simulation for a) 10 MeV, b) 20 MeV, c) 50 MeV, d) 100 MeV, e) 200 MeV and f ) 500 MeV electrons crossing a 0.1 mm Mylar window and propagating for 1.3 m in air. The total simulation box is 0.5 × 0.5 × 2.0m3 .

resistance. The window thickness chosen, 0.1 mm, is actually smaller than what is mechanically feasible, and is used as a lower limit for this parameter. Another option for the window material would be Beryllium, which has a smaller atomic number, but given that this is a hazardous material, and that little advantage is gained by its use, we decided against it. The air distance of 1.3 m was chosen because it matches the largest length an electron would have to travel on air after exiting the vacuum chamber window until it reaches the focal point. Simulations were done for electron energies ranging from 10 MeV to 500 MeV. As we can see, for energies of up to 50 MeV the collective effect of the exit window plus the air results in a large spreading of the electron beam. The initial beam is perfectly collimated, and at the end of the simulation box the beam diameter becomes several centimeters wide, which renders it unsuitable for use with the electron spectrometer. For energies above 100 MeV however this approach appears to be adequate. One option that was less expensive than using a vacuum box was to create an Helium environment for electron propagation. This environment

87

88

Electron Spectrometer System Energy [MeV] 10 20 50 100

Beam Diameter [cm] 12.64 4.59 2.29 0.86

Full divergence [deg] 5.56 2.02 1.01 0.37

Table 4.3: Simulation results for GEANT4 runs.

would be at atmospheric pressure which would greatly simplify the mechanical design of such system. This solution, however, is not applicable because the main problem resides in the exit window. Figure 4.15 presents simulations for a 10 MeV electron beam crossing a number of window/world materials. For simulation c) and d) we used a vacuum window i.e. a virtual window separating the main vacuum chamber from the spectrometer area and as we can see using a Helium environment (fig. 4.15 c) would result in an almost usable situation. However, using a thin mylar window always results in a large beam divergence as shown in fig. 4.15 e) where the simulation was done entirely in vacuum, using a mylar window. We therefore require a vacuum box for operating at lower (< 100MeV) energies. A custom vacuum box was designed for our magnet, based on the fact that the magnet is made of two separable halves, and that the pole pieces include a 20 mm chamfer. The vacuum box is placed between the two parts of the magnet and is fixed into place using simply by reassembling the magnet. The coil supports are replaced by new custom ones that also attach to the vacuum box and serve as supports for this part. For simplicity we designed this box so that it could be built from a single block of non-magnetic stainless steel. No welding or assembly is required except for the entry port, where a standard DN 10 ISO-KF flange with a pipe socket is welded to the box. The vacuum box is then connected to the main vacuum chamber using a standard bellows connector. Vacuum seal is achieved using a teflon ring that is placed between the pole piece chamfer and the vacuum box. Figure 4.16 shows the 3D rendering of the vacuum box model. This vacuum box is currently under construction at a mechanical workshop.

4.3

Detection System

Signal acquisition for the detection system is done differently for the two detectors. The scintillating plate is used either with standard 35 mm panchromatic film or with a CCD camera. When used with film, the detector setup needs

4.3 Detection System

Figure 4.15: GEANT4 simulation for 10 MeV electrons with several window/world materials: a) Mylar/Air, b) Mylar/Helium, c) Vacuum/Air, d) Vacuum/Helium, and e) Mylar/Vacuum. The remaining simulation parameters are the same as in fig. 4.14.

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Figure 4.16: Vacuum box for the spectrometer. magnet is not shown for clarity.

The top half of the

to be prepared in a dark room. We place a the film between two layers of scintillating plate which are cut to the measure of the film. These are then placed inside a black plastic cover which is then sealed on both ends. This plastic cover has a calibration mark that should be aligned with the 90o deflection angle. After the detection, the scintillating assembly (scintillator + film) is taken to a dark room where before removing the film a small puncture is done on the 90o mark. The film is then developed using standard development techniques, and its density is then measured using a calibrated 32 bit scanner. For practical use several scintillating assemblies are prepared so that it is not required to return to the dark room at the end of each shot. When imaged directly by CCD camera, the scintillator is placed along the focal plane without any cover and is imaged using an f/16 aspheric lens system. When using using this setup it is necessary to isolate the scintillator and CCD camera from the rest of the experiment using a thin black plastic cover so that only light coming from the detection of electrons is measured. The data acquisition of the array of scintillating crystals is done using custom designed electronics. Signal processing for this type of detectors is usually done through charge sensitive amplifiers that integrate the current coming

4.3 Detection System from the photodiode. These amplifiers follow the standard Miller integrator configuration [80] and have an extra resistor in parallel with the capacitor for discharging the amplifier and stabilizing the amplifier. The amplifier we designed can be seen on figure 4.17. The output signal from this amplifier, V , will depend on the value of the capacitor C through V = Q/C where Q is the integrated charge, which means that the smallest capacitor will produce the highest signal. A small capacitor (10 pF) is shown on figure 4.17 but in case of saturation of the amplifier larger capacitors can be used. When using small capacitors special attention must be given to the connection between the photodiode and the circuit as the added cable capacitance will reduce signal output. The time constant of this circuit was chosen by adjusting the resistor value so that no oscillation of the output voltage would occur after the main pulse, and that the peak value of the output signal would be linearly proportional the the integrated charge. Note that this requires matching the time constant to the photodiode capacitance. When choosing the operational amplifier for the integrator the key requirement was that this unit would be a very low leakage device. Also, given the short duration of the pulse, a high slew rate operational amplifier is required in other to maximize the dynamic range. We chose the LMC7101 operational amplifier from National Semiconductor [81] for its cost/performance ratio and also because it has very low bias offset (' 12µV) and low noise which makes it a good choice for this type of circuit. The input bias current on this amplifier is ' 100pA sets the minimum detection threshold for the signal coming from the photodiode. For ultra-low noise and high sensitivity the OPA111 from Burr-Brown [82] or the Amptek A225 can be used which have lower leakage currents and higher slew rates; however, this circuit should be able to detect a single electron at 500 KeV using the photodiode and CsI crystals described above, so we decided on the less expensive amplifiers. The output can be measured directly on an oscilloscope provided a follower amplifier with very low input current is added to the output of the charge sensitive amplifier. However, since oscilloscopes are usually limited to four channels each, this would be a very expensive solution. Furthermore, the complete waveform is irrelevant for the final measurement: only the peak value needs to be measured. Taking this in consideration we decided on build our own peak-detector and analog-to-digital converter (ADC) circuit, allowing for greater versatility and lower cost. The output of the ADC can then be sent to a computer for later processing. Figure 4.17 shows this circuit. The peak detector is based on the superdiode [80] configuration. A second diode (D2) is added increase circuit response by avoiding the first amplifier from being in negative saturation. We have also added a 2:1 voltage divider to the output

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Figure 4.17: Signal processing circuit for the data acquisition system. Note: The actual AmpOps used were the LMC2001 which have the same pinout as the LMC7101 but much better performance.

4.3 Detection System

Figure 4.18: Signal processing waveforms for the data acquisition system. Waveforms shown are from SPICE simulations.

amplifier so that this circuit will actually detect twice the peak value, and a MOSFet switch for resetting the peak-detector. To avoid gate-drain leakage on this device, which would charge the peak-detection capacitor, this must be driven at a negative voltage so that an additional operational amplifier was added to convert the digital signal coming from the control electronics. The same amplifiers that where used for the charge sensitive amplifier were used here. The ADC is the 12 bit monolithic ADS7818 unit from Burr-Brown [83] with internal voltage reference and a 1 mV resolution. Using these parameters the full dynamic range for the charge sensitive amplifer will then be 0 - 2.048 V. This ADC also features a sample & hold unit, and serial data output. To avoid interference and added cable capacitance we have decided to do all analog signal processing at the detectors, and have only digital signals on the cables. This meant designing the electronics system so that the chargesensitive amplifier, peak-detector and ADC would fit on a small printed circuit board. Figure 4.19 shows the first prototype built for this purpose, which includes some test leads and additional amplifiers so that detailed monitoring

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Figure 4.19: Detector electronics prototype.

can be done on this unit. The photodiode that is attached to the CsI crystal is then soldered directly to this printed circuit board and the whole circuit is then mounted inside a thick metal box for electrical shielding. A 5 × 5 mm opening is added to this box so that electrons can reach the CsI crystal. All external electrical connections, including serial data output, are done using a shielded RJ45 plug and S-FTP Cat. 6 cable. We have also created a triggering system for the pre-amplifiers. This unit can be used both for normal operation, where a digital trigger from the main laser system will be used, and calibration, where the preamplifier unit in use will be self-triggered. Figure 4.20 shows this circuit. A switch allows you to choose between normal and calibration operations. When being used in the calibration mode, the output of the charge sensitive amplifier is processed through a follower amplifier based on the same operational amplifier used earlier, and is then processed by a signal discriminator, whose level is also set by a trim-potentiometer. The discriminator signal is then converted to a digital TTL level signal using a standard MOSFET switch configuration. This signal will than trigger a delay generator based on a pair of standard RC monostable oscillators. The pre-amplifiers do not require high precision triggering, so this simple delay generator will suffice, provided we use high quality capacitors. The delays can be set through a pair of trim-potentiometers. The output of the trigger unit is then connected to the pre-amplifiers. When using more than 8 detectors, additional digital buffers must be added to reduce the fano factor.

4.4 Computer Interface

Figure 4.20: Trigger circuit for testing and calibration.

The added delay of these buffers is negligible for normal operation.

4.4

Computer Interface

The ADC unit selected outputs the converted data through a serial connection immediately after conversion. This data then needs to stored and sent to a computer for further processing. For better operation we also decided that this circuit, instead of relying on an external trigger, should detect when any of its inputs is sending data and store it in the corresponding buffer. This circuit will also be responsible for the clock generation for all pre-amplifiers, which removes any synchronization problems between these units. Given the high complexity of such circuit, as well as large storage necessity, we decided on building it using programmable logic, which greatly simplifies development, implementation, and expansion. The chosen devices for this belong to the Xilinx 70XX CPLD (complex programmable logic device) family [84], that feature high speed and gate density at a low cost. The initial implementation was done on a 70148 device, which features 148 logic units, and that can be found on the XS48 prototyping board. This unit has enough capacity to handle

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Figure 4.21: Data acquisition and Computer Interface circuit.

4.4 Computer Interface

Figure 4.22: Data acquisition and computer interface prototype.

4 detectors using the circuit shown on figure 4.21 (Only main modules are shown). This design can be expanded up to 2048 detectors without changing the computer interface, but it will require a much larger CPLD. The modules U4, U7, U8 and U9 detect when the ADC on the corresponding detector starts sending data and store it internally. Module U10 can be programmed through the computer interface to select which of these will be enabled for output. To minimize gate usage, U4, U7, U8 and U9 are connected in a cascading configuration which means some delay occurs before the data can be read by the computer interface. All these modules were programmed using the ABEL hardware description language [84]. This circuit also features a ”busy” and a ”new data” output that indicate that data is being read from the detectors and that new data is ready for acquisition respectively. The computer interface hardware is comprised of an ActiveWire USB interface card [85]. This is a low cost 20 bit parallel I/O USB card that allows for easy integration with our detection system. Figure 4.22 shows the full data aquisition/computer interface unit. The software layer for data acquisition was developed in LabView [86] and is based on the custom set of drivers provided with the USB board. The only user setting required from the user is to specify the number of detectors being used. This software also allows for the generation of the energy distribution plots and includes multi-channel

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98

Electron Spectrometer System analysis software for the calibration of the detection system. Calibration of the detection system is done using a set of low activity radioactive sources that produce beta particles. The scintillating plate is calibrated by exposing a section to the radioactive source for a long period and imaging it with photographic film. Using the measured activity of the radioactive source we then measure the total photon emission for this period of time and determine the photon yield. For the scintillating crystals, we use a similar procedure, but instead of simply integrating over all the beta particles, we use the multi-channel software routines to determine the central photon yield. This procedure has the advantage of calibrating the full crystal - photodiode - pre-amplifier - peak detector setup, resulting in a single calibrating curve for each detector. Whenever possible, the scintillating plates should be calibrated against the crystal detectors, since the latter are much more accurate. The calibrating data obtained can then be extrapolated to the full energy range by converting the energy loss to higher energy electrons.

Chapter 5 PIC Simulation of Plasmas

5.1

Introduction

Based on the highly nonlinear and kinetic processes that occur during highintensity laser-plasma interactions, we use particle-in-cell (PIC) codes [87, 88] for the modeling of these physical problems. In these codes the full set of Maxwell’s equations are solved on a grid using currents and charge densities calculated by weighting discrete particles onto the grid. Each particle is pushed to a new position and momentum via self-consistently calculated fields. Therefore, to the extent that quantum mechanical effects can be neglected, these codes make no physics approximations and are ideally suited for studying complex systems with many degrees of freedom.

5.2

Numerical Simulation of Plasmas

While attempting to simulate plasma behavior we may first be tempted to simulate the interaction of all the particles directly, using what is know has the Particle-Particle method. A quick estimate of the number of calculations necessary per time step is straightforward. The time step would have two major parts: calculating the forces acting on the particles and pushing the particles. For a Np number of particles the calculation of the forces we would first clear the force accumulator on each particle, which would use 3Np operations. Then, assuming Coulomb interactions in 3D we would require for each particle pair: 3 operations for xj − xi , 8 operations for |xj − xi |3 , 3 operations for the force between the two particles Fij , and 6 more operation to accumulate Fi and Fj .

100

PIC Simulation of Plasmas Since we have Np (Np − 1)/2 particle pairs this gives a total of 10Np (Np − 1) operations. For pushing the particles we need to first update the velocities and then update the positions which gives a total of 6Np operations. The total number of operations needed on one time step would then be 10Np2 − Np , and therefore scales with N 2 . Using a state-of-the art 1 GFlop/s processor, the loop time for 5 million particles (the number of particles in a few Debye spheres for confined fusion experiments), we get about 2.9 days. A typical run would have about 5000 time steps and would take about 40 years. Although large scale supercomputers in the TeraFlop/s scale could make such simulations feasible, we clearly require a different method for the numerical simulation of plasmas One such method is the so called Particle-Mesh method. In this method forces are exchanged between the particles through fields, in our case, the electromagnetic field. This field is represented approximately by a regular array of mesh points, and field values at the particle positions are obtained by interpolating on the array of mesh-defined values. Mesh-defined densities for the calculation of the field are obtained by the opposite process of depositing the particle attributes to nearby mesh points. The time step loop for the particle-mesh method differs only from the previous method in the calculation of the forces. This calculation is a three-step process: i) Deposit charge on the mesh; ii) Solve Poisson’s equation on the mesh to get the electric field; and iii) Interpolate electric field at the particle positions and calculate the forces acting on each particle. Steps i) and iii) have a number of operations proportional to the number of particles Np , and step iii) depends on the number of mesh points NM . The exact number of operations depends on the particular form of PM scheme implemented. Typical values [89] give 20Np operations for steps i) and iii) and 5NM log2 NM operations for step ii). The loop time for the same 5 million particles, using a 643 mesh, would then be only 125ms, and a 5000 step simulation would take about 11 minutes. (Note that these two estimates are about 10 - 100 times faster than actual simulation times because a) real codes usually achieve only a fraction of the peak processing power and b) there are a number of additional operations - like iterating through all the particles, for example - that are unaccounted for). Still, 5 million particles only represent a small fraction of the total number of particles that exist in relevant problems. A typical laser produced plasma of 1018 e− /cm3 has about 100 times that many particles in a 10µm sphere, and usually the simulation of a few hundred µm is required to model experiments. Except for a few specific problems (see for example [90]) a literal simulation of these problems is still beyond the practical reach of present day computing systems. For this reason we choose to simulate only a fraction

5.3 PIC Codes

101

of the total plasma particles and view each individual simulation particle as representing many particles in the real plasma (a superparticle). This makes good physical sense; we are generally interested in the collective behavior of the plasma and not in the time evolution of each individual particle. For collisionless plasmas such as the ones present in laser-plasma experiments the collective effects of the plasma dominate over individual collisions so an equivalent distribution of superparticles will produce essentially the same results. Furthermore the use of a finite mesh for describing the fields already smooths out forces at lengths smaller than half grid cell, which leads to considering these superparticles as finite-sized clouds of plasma particles, with their position and velocity being the position of the center of mass and average velocity of these clouds. The general drawback of this technique is that of increased noise, because although superparticles have the same charge-to-mass ratios has laboratory particles they have much larger charge and masses. Also, having less particles mean a less accurate statistical description, but again this is usually a good trade-off.

5.3

PIC Codes

The simulation method we use for the modeling of laser-plasma interactions is the Particle-In-Cell method, or PIC. This is a subset of the Particle-Mesh method described above and has successfully been used for the modeling of plasmas since its initial development through the pioneering work of J. Dawson and O. Buneman in the late 1950’s and early 1960’s. Several variations of the PIC method exist [89]. We present here a short review of the fundamental physics and algorithms used in this field. PIC codes are very basic models in the sense that no assumption other than the validity of the particle-mesh technique is done. These codes rely on solving the fundamental physical equations governing particle motion and electromagnetic field evolution. The key equations describing our system are the Lorentz equation for the motion of the particles (in cgs units): v p˙ = q E + × B c

(5.1)

where p is the particle momentum, v is the particle velocity, q is the particle charge, c is the speed of light, and E and B are the electric and magnetic field, respectively. Maxwell’s equations for the electromagnetic field are (again in cgs units):

102

PIC Simulation of Plasmas

∇ · E = 4πρ

∇×B=

1 ∂E 4π + j c ∂t c

−∇ × E =

1 ∂B c ∂t

∇·B=0

(5.2)

(5.3)

(5.4)

(5.5)

where ρ and j are charge density and current density respectively. For modeling laser-plasma interactions we require a code that can resolve highly relativistic electron velocities, so we require a fully electromagnetic code. For advancing the fields we then use only Ampere’s equation (5.3) and Faraday’s equation (5.4) rewritten in the following manner: ∂E = 4πj − c∇ × B ∂t

(5.6)

∂B = −c∇ × E ∂t

(5.7)

Note that only the current density j is required for advancing the fields. The rotational operator is replaced by a finite-difference approximation on the mesh, and E, B and j are defined on shifted meshes for higher accuracy. (In 1D this can be viewed as defining E and j on points xi , and B on points xi+ 1 ). 2 In the beginning of the time integration the magnetic and electric fields are centered at time tn and the current density is centered at time n + 12 . The time integration is done in three steps: i) we first advance the magnetic field by 1 half time step, Bn → Bn+ 2 , then ii) we advance the electric field by a full time step, En → En+1 , and finally iii) we advance the magnetic field by another 1 time step Bn+ 2 → Bn+1 . This integration procedure allows for second order accuracy. The equation (5.1) is solved using the so called Boris pusher [88, 91], using the field values interpolated a the particle initial position xn . This is also a multi-step method for increased accuracy and a detailed derivation can be found in the references presented. For the relativistic generalization of (5.1) we use the generalized velocity, u ≡ γv, where γ is the relativistic Lorentz factor. Equation (5.1) becomes:

5.3 PIC Codes

103

q u˙ = m

1u E+ ×B cγ

(5.8)

Where m is the particle mass. While advancing one full time step, ∆t we 1 start from un− 2 , and use the time centered fields En and Bn . We completely separate the effects of the electric and magnetic forces and do four steps: i) add half the electric impulse to obtain u0 , ii) rotate u0 with half the magnetic impulse to get u00 , iii) rotate u0 with the full magnetic impulse using u00 , and iv) add the remaining half of the electric impulse. Explicitly: 1

u0 = un− 2 +

q ∆t n E m 2

γ n = 1 + u02

t=

21

q ∆t Bn m 2 γn

u00 = u0 + u0 × t

s=

2t 1 + t2

u000 = u0 + u00 × s 1

un+ 2 = u000 +

q ∆t n E m 2

(5.9)

The updated momentum is then used to update the particle position through: γ

n+ 12

h

= 1 + (u

n+ 12 2

)

i 12

1

n+1

x

n

=x +

un+ 2 1

γ n+ 2

∆t

(5.10)

After the position update we deposit the particle current using the line path defined by xn+1 − xn . We split this path into the segments lying inside each

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Figure 5.1: Loop cycle for the electromagnetic PIC algorithm used. The particles are numbered i = 1, 2, · · · , NP and the grid indexes are j, which becomes a vector in 2 and 3 dimensions.

individual cell, and then deposit each segment to the nearest grid points. This method allows for greater spatial accuracy and result in the deposition of 1 current centered at time n + 12 : jn+ 2 , which is necessary for the above field calculations. Figure 5.1 shows the loop cycle for our PIC algorithm. Beginning with a set of particle positions and velocities, and initial electric and magnetic fields, we first interpolate the field values at the particle positions. Then, using the interpolated fields, we integrate the equations of motion of the particles. We then proceed to depositing the current density values on the grid and finally we use these values for integrating the field equations. Note that the algorithm used requires that we supply a self-consistent initial condition for the electric and magnetic field. If we do not do so the code will take a few tens of time steps while evolving to a self-consistent situation and only then will begin to produce physically meaningful values. Another thing worth mentioning is that PIC simulations are usually done in normalized units, which has two distinct advantages: i) multiplication by several constants (like me , e and c, for example) is avoided, resulting in a significant performance increase and ii) by expressing the simulation quantities in terms of the fundamental plasma quantities the results are general and not

5.3 PIC Codes

105

bound to some specific units we may choose. In our case we chose to normalize our quantities to ωp , me , c and e, the electron plasma frequency, the electron rest mass, the speed of light and the electron charge, respectively. For the position, linear momentum, electric field and magnetic field we get: ωp x c

(5.11)

p γv u = = msp c c c

(5.12)

x0 =

p0 =

E0 = e

c/ωp E me c2

(5.13)

B0 = e

c/ωp B me c2

(5.14)

where msp is the mass of the species being considered. In this situation γ can be calculated as γ = (1 + p02 )1/2 . In practical units the physical quantities will relate to simulation quantities as: x[cm] = 2.998 × 1010 x0 ωp−1 [rad/s] (5.15) 0

6

x[cm] = 0.532 × 10 x

−1/2 n0 [cm−3 ]

p[g cm/s] = 2.731 × 10−17

msp 0 p me

(5.16)

E[GV/cm] = 1.704 × 10−14 E0 ωp [rad/s] (5.17) −10

E[GV/cm] = 9.613 × 10

E

0

1/2 n0 [cm−3 ]

B[gauss] = 5.681 × 10−8 B0 ωp [rad/s] (5.18) −3

B[gauss] = 3.204 × 10

B

0

1/2 n0 [cm−3 ]

Also note that for high kinetic energies, p msp c, the relativistic energy is reduced to W ' msp c2 p0 , where msp c2 is the rest mass energy for the species being considered.

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5.4

Osiris

Achieving the goal of one to one, two and three dimensional modeling of laboratory experiments and astrophysical scenarios, requires state-of-the-art computing systems. The rapid increase in computing power and memory of these systems that has resulted from parallel computing has been at the expense of having to use more complicated computer architectures. In order to take full advantage of these developments it has become necessary to use more complex simulation codes. The added complexity arises for two reasons. One reason is that the realistic simulation of a problem requires a larger number of more complex algorithms interacting with each other than in a simulation of a rather simple model system. For example, initializing an arbitrary number of lasers or particle beams in 3D on a parallel computer is a much more difficult problem than initializing one beam in 1D or 2D on a single processor. The other reason is that the computer systems, e.g., memory management, threads, operating systems, are more complex and as a result the performance obtained from them can dramatically differ depending on the code strategy. Parallelized codes that handle the problems of parallel communications and parallel IO are examples of this. The best way to deal with this increased complexity is through an object-oriented programming style that divides the code and data structures into independent classes of objects. This programming style maximizes code reusability, reliability, and portability.

The goal of this code development project was to create a code that breaks up the large problem of a simulation into a set of essentially independent smaller problems that can be solved separately from each other. This allows individuals in a code development team to work independently. Object oriented programming achieves this by handling different aspects of the problem in different modules (classes) that communicate through well-defined interfaces.

This effort resulted in a new framework called OSIRIS, which is a fully parallelized, fully implicit, fully relativistic, and fully object-oriented PIC code, for modeling intense beam plasma interactions. Details of the status of OSIRIS several years ago can be found in [34]. The evolution of OSIRIS is due to the combined efforts of many people. In this paper we describe some of this evolution and the current status of OSIRIS.

5.4 Osiris

5.4.1

Development

The programming language chosen for this purpose was Fortran 90, mainly because it allows us to more easily integrate already available Fortran algorithms into this new framework that we call OSIRIS. We have also developed techniques where the Fortran 90 modules can interface to C and C++ libraries, allowing for the inclusion of other libraries that do not supply a Fortran interface. Although Fortran 90 is not an object-oriented language per se, object-oriented concepts can be easily implemented [92, 93, 94] by the use of polymorphic structures and function overloading. In developing OSIRIS we followed a number of general principles in order to assure that we were building a framework that would achieve the goals stated above. In this sense all real physical quantities have a corresponding object in the code making the physics being modeled clear and therefore easier to maintain, modify and extend. Also, the code is written in a way such that it is largely independent from the dimensionality or the coordinate system used, with much of the code reused in all simulation modes. Regarding the parallelization issues, the overall structure allows for an arbitrary domain decomposition in any of the spatial coordinates of the simulation, with an effective load balancing of the problems in study. The input file defines only the global physical problem to be simulated and the domain decomposition desired, so that the user can focus on the actual physical problem and does not need to worry about parallelization details. Furthermore, all classes and objects refer to a single node (with the obvious exception of the object responsible for maintaining the global parallel information), which can be realized by treating all communication between physical objects as a boundary value problem, as described below. This allows for new algorithms to be incorporated into the code, without a deep understanding of the underlying communication structure.

5.4.2

Design

Object-Oriented Hierarchy Figure 5.2 shows the class hierarchy of OSIRIS. The main physical objects used are particle objects, electromagnetic field objects, and current field objects. The particle object is an aggregate of an arbitrary number of particle species objects. The most important support classes are the variabledimensionality-field class, which is used by the electromagnetic and current

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Figure 5.2: Osiris main class hierarchy.

field classes and encapsulates many aspects of the dimensionality of a simulation, and the domain-decomposition class, which handles all communication between nodes. Benchmarking of the code has indicated that the additional overhead from using an object oriented framework in Fortran 90 leads to only a 12% slowdown in speed. Parallelization The parallelization of the code is done for distributed memory systems, and it is based on the MPI message-passing interface [95]. We parallelize our algorithms by decomposing the simulation space evenly across the available computational nodes. This decomposition is done by dividing each spatial direction of the simulation into a fixed number of segments (N1 , N2 , N3 ). The total number of nodes being used is therefore the product of these three quantities (or two quantities for the 2D simulations). The communication pattern follows the usual procedure for a particlemesh code [96]. The grid quantities are updated by exchanging (electric and magnetic fields) or adding (currents) the ghost cells between neighboring nodes. As for the particles, those crossing the node boundary are counted and copied to a temporary buffer. Two messages are then sent, the first with the number of particles, and the second with the actual particle data. This strategy allows for not setting an a priori limit on the number of particles being sent to another node, while maintaining a reduced number of messages. Because most of the

5.4 Osiris message are small, we are generally limited by the latency of the network being used. To overcome this whenever possible the messages being sent are packed into a single one, achieving in many cases twice the performance. We also took great care in encapsulating all parallelization as boundary value routines. In this sense, the boundary conditions that each physical object has can either be some numerical implementation of the usual boundary conditions in these problems or simply a boundary to another node. The base classes that define grid and particle quantities already include the necessary routines to handle the latter case, greatly simplifying the implementation of new quantities and algorithms.

Encapsulation of System Dependent Code For ease in porting the code to different architectures, all code that is machine dependent is encapsulated in the system module. At present we have different versions of this module for running on the Cray T3E, the IBM SP, and for Macintosh clusters, running on both MacOS 9 (MacMPI [97]) and MacOS X (LAM/MPI [98]) clusters. The latter is actually a Fortran module that interfaces with a POSIX compliant C module and should therefore compile on most UNIX systems, allowing the code to run on PC-based (Beowulf) clusters. The MPI library has also been implemented on all these systems requiring no additional effort.

Code Flow Figure 5.3 shows the flow of a single time step on a typical OSIRIS run. It closely follows the typical PIC cycle [88]. The loop begins by executing the diagnostic routines selected (diagnostics). It follows by pushing the particles using the updated values for the fields and depositing the current (advance deposit). After this step, the code updates the boundaries for particles and currents, communicating with neighboring nodes if necessary. A smoothing of the deposited currents, according to the specified input file, follows this step. Finally, the new values of the electric and magnetic field are calculated using the smoothed current values, and its boundaries are updated, again communicating with neighboring nodes, if necessary. If requested, at the end of each loop, the code will write restart information, allowing the simulation to be restarted later on at this time step.

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Figure 5.3: A typical cycle, one time step, in an OSIRIS 2 node run. The arrows show the direction of communication between nodes.

5.4.3

OSIRIS Framework

The code is fully relativistic and it presently uses either the charge-conserving current deposition schemes from ISIS [99] or TRISTAN [100]. We have primarily adopted the charge-conserving current deposition algorithms because they allow the field solve to be done locally, i.e., there is no need for a Poisson solve. The code uses the Boris scheme to push the particles, and the field solve is done locally using a finite difference solver for the electric and magnetic fields in both space and time. In its present state the code contains algorithms for 2D and 3D simulations in Cartesian coordinates and for 2D simulations in azimuthally symmetric cylindrical coordinates, all of which with 3 components in velocity (i.e. both 2D modes are indeed 2 21 D or 2D3V algorithms). The loading of particles is done by distributing the particles evenly on the cell, and varying the individual charge of each particle according to the density profile stipulated. Below a given threshold no particles are loaded. The required profile can be specified by a set of multiplying piecewise linear functions and/or by specifying Gaussian profiles. The initial velocities of the particles are set according to the specified thermal distribution and fluid velocity. The code also allows for the definition of constant external electric and magnetic fields. The boundary conditions we have implemented in OSIRIS are: conducting, and Lindmann open-space boundaries for the fields [101], and absorbing, reflective, and thermal bath boundaries for the particles (the later consists of reinjecting any particle leaving the box with a velocity taken from a ther-

5.4 Osiris mal distribution). Furthermore, periodic boundary conditions for fields and particles are also implemented. This code also has a moving window, which makes it ideal for modeling high-intensity beam plasma interactions where the beam is typically much shorter than the interaction length. In this situation the simulation is done in the laboratory reference frame. Simulation data is shifted in the direction opposite to the motion of the window whenever nc∆t > ∆x where n is the first integer for which this inequality is satisfied. Since this window moves at the speed of light in vacuum no other operations are required. The shifting of data is done locally on each node, and boundaries are updated using the standard routines developed for handling boundaries, thus taking care of moving data between adjacent nodes. The particles leaving the box from the back are removed from the simulation and the new clean cells in the front of the box are initialized as described above. OSIRIS also incorporates the ability to launch EM waves into the simulation, either by initializing the EM field of the simulation box accordingly, or by injecting them from the simulation boundaries (e.g. antennas). Moreover, a subcycling scheme [102] for heavier particles has been implemented, where the heavier species are only pushed after a number of time steps using the averaged fields over these time steps, thus significantly decreasing the total loop time. We have also implemented particle sorting routines that allow for better use of CPU cache memory, resulting in performance gains of up to 40%. A great deal of effort was also devoted to the development of diagnostics for this code that goes beyond the simple dumps of simulation quantities. For all the grid quantities envelope and boxcar averaged diagnostics are implemented; for the EM fields we implemented energy diagnostics, both spatially integrated and resolved; and for the particles phase space diagnostics, total energy and energy distribution function, and accelerated particle selection are available. The output data uses the HDF [103] file format. This is a standard, platform independent, self-contained file format, which gives us the possibility of adding extra information to the file, like data units and iteration number, greatly simplifying the data analysis process.

5.4.4

Performance

The code has been successfully used in the modeling of several problems in the field of plasma based accelerators, and has been run on a number of architectures. Table 5.1 shows the typical push times on three machines, two supercomputers and one computer cluster.

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PIC Simulation of Plasmas Machine IBM SP Cray T3E-900 EP2 Cluster

2D push time 3.82 5.77 4.96

3D push time 7.32 11.2 9.82

Table 5.1: Typical push time for three machines, in two and three dimensions. Values are in µs/particle × node.

We have also established the energy conservation of the code to be better than 1 part in 105 . This test was done in a simulation where we simply let a warm plasma evolve in time; in conditions where we inject high energy fluxes into the simulation (laser or beam plasma interaction runs) the results are better. Regarding the parallelization of the code, extensive testing was done on the EP2 cluster [104] at the IST in Lisbon, Portugal. We get very high efficiency, (above 91% in any condition), proving that the parallelization strategy is appropriate. Also note that this is a computer cluster running a 100 Mbit/s network, and that the efficiency on machines such as the Cray T3E is even better

5.5

Visualization and Data-Analysis Infrastructure

It is not an exaggeration to say that visualization is a major part of a parallel computing lab. The data sets from current simulations are both large and complex. These sets can have up to five free parameters for field data: three spatial dimensions, time and the different components (i.e., Ex , Ey , and Ez ). For particles, phase space has seven dimensions: three for space, three for momentum and one for time. Plots of y versus x are simply not enough. Sophisticated graphics are needed to present so much data in a manner that is easily accessible and understandable. We developed a visualization and analysis infrastructure [105] based on IDL (Interactive Data Language). IDL is a 4GL language, with sophisticated graphics capabilities, and it is widely used in areas such as Atmospheric Sciences and Astronomy. It is also available on several platforms and supported in a number of systems, ranging from Solaris to the MacOS. While developing this infrastructure we tried simplifying the visualization and data analysis as much as possible, making it user-friendly, automating as much of the process as possible, developing routines to batch process

5.6 EP2 Cluster

Figure 5.4: Force field acting on the 30 GeV SLAC beam inside a plasma column.

large sets of data and minimizing the effort of creating presentation quality graphics. We implemented a full set of visualization routines for one, two and three-dimensional scalar data and for two and three dimensional vector data. These include automatic scaling, dynamic zooming and axis scaling, integration of analysis tools, animation tools, and can be used either in batch mode or in interactive mode. We have also developed a comprehensive set of analysis routines that include scalar and vector algebra for single or multiple datasets, boxcar averaging, spectral analysis and spectral filtering, k-space distribution function, envelope analysis, mass centroid analysis and local peak tools. One example of the analysis and visualization of a three-dimensional modeling of a plasma accelerator is presented on figure 5.4. This is a oneto-one modeling of the E-157 Experiment [106] done at the Stanford Linear Accelerator Center, where a 30 GeV beam is accelerated by 1 GeV. The figure shows the Lorentz forces acting on the laser beam e.g. E + z × B, where z is the beam propagation direction, and we can clearly identify the focusing /defocusing and accelerating/decelerating regions. This visualization work was awarded the Oscar Buneman award for best visualization at the 2000 International Conference for the Numerical Simulation of plasmas.

5.6

EP2 Cluster

All simulation work done on this thesis was done at the EP2 cluster at IST and extensive work was done on its development. The use of a computer cluster instead of a traditional supercomputer for the problem sizes we are running actually resulted on a faster turn around time because we had no waiting queues

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PIC Simulation of Plasmas Processor OS Comm. Libs. NIC Network RAM Storage Peak Performance

16 Dual Power Mac G4/450, 2x G4 PPC/450 Mhz Mac OS X 10.1.2 LAM/MPI 6.5.6 (including MPE and ROM-IO) 1Gbit/s Ethernet Cards Asanté Intracore 8000 10/100 Mbit/s switch 1.125 Gbyte/node, 18 Gbyte total 30 Gbyte/node, 480 Gbyte total 57.6 GFlop/s

Table 5.2: EP2 Cluster characteristics.

to deal with and the several Gigabytes of data produced were already available for post processing locally, not requiring a lengthy download from a remote computing environment. The EP2 cluster is based on the Dual Power Mac G4 computer from Apple [107] and on the Appleseed paradigm [97]. The choice of Macintosh computers over the usual Intel x86 Beowulf architecture was mainly related to the low installation and maintenance requirements that characterize Macintosh systems. As for performance, the Power PC G4 processor, from Apple, Motorola and IBM, is capable of a sustained performance of over one Gigaflop, and presents excellent characteristics for scientific computing. Furthermore, Apple’s new operating system, Mac OS X, has a UNIX core which allows us to use most computational libraries available, and, together with hardware integration done at the manufacturer, results in superior reliability and stablity. Table 5.2 presents the general EP2 Cluster characteristics. The network backbone is provided by a Asanté Intracore 8000 10/100 Mbit/s switch [108]. This switch was chosen for its performance and scalability, providing a fully meshed network and assuring a 100 Mbit/s network connection between any two ports. Power supply is done through two Pulsar EXtreme 3000 UPS units from MGE [109]. The cluster also includes two 17” CRT monitors and two 8 port KVM-USB switches from Dr. Bott [110] for installation and maintenance purposes. The cluster room had to be fitted with an appropriate air conditioning system, given the high power dissipation from the 16 computers (∼ 6 KW). Figure 5.5 shows the EP2 Cluster without the KVM switches One critical aspect of computer clusters for parallel computing is their network performance. Careful benchmarking of the EP2 cluster was done using the Ping-Pong program developed by V. Decyk and co-workers at UCLA [97]. This programs tests both half-duplex and full-duplex communication performance for number of packet sizes. Results are shown in figure 5.6 using LAM-MPI as the message passing protocol [98]. (The results with MacMPI

5.6 EP2 Cluster

Figure 5.5: The EP2 Cluster at IST. The cluster has 16 dual Macintosh G4’s, an Asant´ e Intracore 8000 10/100 network switch (top left) and two 17” Monitors for maintenance and installation purposes.

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Figure 5.6: Network performance of the EP2 cluster with packet sizes ranging from 1 byte to 2 Megabytes, using half-duplex (Ping) and fullduplex (Swap) communications.

[97] are slightly slower for intermediate package sizes). As we can see for small packages the latency time dominates and we get poor performance. For large packets however the results are very good, reaching the theoretical limits on a 100 Mbit network. This benchmark also analyzes the stability of the network, by repeating the same measurement several times to obtain its statistical behavior. The error bars on the plot show that we have excellent stability, even in the full-duplex mode. We have also done extensive benchmarking of OSIRIS running on the EP2 cluster. Two sets of tests were done. The first set tests parallelization efficiency with a problem of increased granularity. The reference problem is the 2D simulation of the collision of two electron clouds moving perpendicularly to the simulation plane. The simulation is done on a 1024 × 1024 grid, with two particle species, with 4 million particles per species (4 particles / species / cell), and we run it for 100 time steps. After running this problem on a single cpu we divide the problem into 2, 4, 8, and 16 nodes, using 1 cpu per computer, and 2, 4, 8, 16, and 32 nodes, using 2 cpu’s per computer. Results are shown in figure 5.7. The second set tests the scalability efficiency of the cluster measured as the time it takes to run a problem proportionally large to the number of nodes. The reference problem is the same as in the previous benchmarks, but with a 1024 × 512 grid, and again using 4 particles / species

5.6 EP2 Cluster

Figure 5.7: Parallel performance of OSIRIS 2D running on the EP2 cluster for the granularity benchmarks.

/ cell. The simulations were run on the same number of nodes as before, for both 1 and 2 cpu’s per computer, with a problem size of (#Nodes×1024)×512 for 100 time steps. Results are shown in figure 5.8, normalized to the 1 node run. As we can see we get excellent results on both tests. The parallel efficiency for the first set of tests remains above 93% for the most of the situations and has a minimum of 85% using 32 nodes. The reason for this performance degradation is that problem size is now very small and communication time begins to be on the same order as computation time, but it remains at a very high value. We can also observe the effect of using 2 cpu’s per machine on cluster performance. The performance degradation on this case (which is related with sharing memory bandwidth, for example) is negligible, and dual cpu’s configurations should be used by default. Regarding the scalability tests we also get very good results. The reference problem took 48 minutes to run. For 1 cpu per computer, and after an initial small drop in performance, when beginning to run the problem on multiple nodes, cluster performance stabilizes and varies less 0.5% throughout all the test, remaining at a scalability efficiency of about 98%. For 2 cpu’s per computer we again observe a slight performance degradation, but still remain above 94% throughout all the test. This means that running a simulation with 128 million particles on 32 nodes took only 3 more minutes to do (51 minutes) than running a 4 million particle simulation on 1 nodes.

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Figure 5.8: Parallel performance of OSIRIS 2D running on the EP2 cluster for the scalability benchmarks, normalized to the 1 node run. Note that the vertical axes scales from 90% to 100%.

Chapter 6 Simulation of Laser Acceleration Experiments at L2I In this chapter we present the numerical modeling of laser plasma accelerator experiments under the L2I laser laboratory conditions. These simulations are a one to one modeling of the of the L2I laboratory conditions and, besides the 2D PIC model used, make no physical approximations and the simulations are run with the exact time and length dimensions of the experiment. All simulation work presented in this chapter was done using the OSIRIS framework on the EP2 cluster, as described in the previous chapter. The main goal of this simulation work is to explore the possibilities that the L2I laboratory now provides in the field of laser-plasma accelerator experiments, in order to find an estimate of the best experimental parameters. A detailed study of the laser wakefield accelerator configuration and a parametric analysis of the channeled laser plasma accelerator are presented for the existing laboratory conditions. Furthermore, it provides a general guide for the design of channeled accelerator experiments.

6.1

The L2I Laboratory

The Laboratory for Intense Lasers (L2I) is the facility of the Grupo de Lasers e Plasmas dedicated to the research in the fields of laser plasma and laser matter interaction. It is installed in a 120m2 Class 10000 clean room located in the main Instituto Superior Técnico campus in Lisbon, Portugal. This laboratory includes a high power laser facility producing Terawatt-level (1012 W) light

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Simulation of Laser Acceleration Experiments at L2I

Figure 6.1: General view of the L2I laboratory showing the laser system in the foreground and the interaction area in the background.

pulses using the chirped-pulse amplification concept, a target area, control and acquisition equipment and relevant diagnostics. Figure 6.1 shows a general view of the laboratory. Our main laser system consists of a Ti:sapphire - Nd:glass chirped pulse amplification chain delivering pulses in the sub-picosecond range. The installation of the first version of the L2I CPA laser system was finished in early 2000. At this stage, the laser pulses produced had 2.6 ps FWHM duration and an energy of about 1 J. The use of a 30 meter glass optical fiber after the oscillator allowed the reduction of the pulse duration to 800 fs increasing the peak power beyond the Terawatt level. The first optical field ionized plasmas were produced shortly after this upgrade was completed, in middle 2000, and the first experiments in the target area began in early 2001. In January 2002 the laser system oscillator was upgraded to a new Ti:Sapphire laser, increasing the final pulse contrast and quality, reducing the pulse duration to less than 200 fs and increasing the peak power to above 4 TW. The present laser parameters are shown in table 6.1. Further details can be found in references [2] and [111]. The target area of the L2I laboratory comprises a large vacuum chamber (1.2 m in diameter) installed in a vibration isolating stand. The beam delivery system includes probe beams and delay lines and allows for several geometries using 1 to 3 beams from the main laser system to be used for the experiments. Solid targets and gas targets [112] can be used for laser matter and laser

6.1 The L2I Laboratory λ0 1.053 µm ∼ 155 fs τL a0 3.23 5.1 µm w0 (radius) I 1.27 · 1019 W/cm2 Table 6.1: Current L2I laser paramters. a0 , w0 , and I values are shown for the 30 cm parabolic mirror configuration.

plasma interaction experiments inside this chamber. Energy and beam quality diagnostics are available, as well as autocorrelators, optical spectrometers and other relevant diagnostics. This target area includes a centralized triggering system allowing for easy integration of any diagnostics, as well as a specialized triggering system for CCD cameras. It also includes two optical tables for aditional diagnotic setups. Further detail about the L2I target area can be found in [28].

One of the main problems in laser plasma accelerators is that of laser diffraction. This problem is usually addressed by using some form of guiding to avoid this behavior, usually through a pre-formed tailored density plasma channel that would propagate the laser pulse through many diffraction lengths. Discharge based plasma channels are considered to be the most promising method to produce long plasma channels because they do not require expensive laser pulses and complicated optical system. In middle 2001 the development of a laser triggered discharge plasma channel generator began at L2I to fit the laboratory with the necessary tools to study such systems. The main advantage of the system being developed is that the discharge is triggered by a laser pulse (a small portion of the main laser pulse is diverted for this purpose), thus avoiding any jitter in the channel formation relatively to the main laser pulse we intend to guide. This system consists mainly of two electrodes with a stamped cone and a hole on the center placed at a variable distance (the required channel length) in a custom vacuum chamber filled with a low pressure hydrogen or helium gas. An high voltage is applied to these electrodes, just below the breakdown limit. When the triggering pulse hits the neutral gas near the cathode it ionizes it. The produced (free) electrons are then accelerated by the applied electrical field, ionizing the remaining neutral gas in the channel region and triggering the discharge. Figure 6.2 shows a shadowgraphy image of one of these channels. Further details on this system can be found in [28].

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Figure 6.2: Shadowgraphy image of a laser triggered discharge plasma channel produced at the L2I laboratory. Courtesy of N. C. Lopes.

6.2

Laser Wakefield Accelerator

The laser wakefield accelerator decribed in chapter 1 is the simplest configuration of a laser plasma accelerator. This configuration requires that the laser pulse length approximately matches the plasma wavelength i.e. cτL ≈ λp so that it will resonantly excite a plasma wave as described in [1]. For a laser pulse duration of 150 fs this requires a plasma density of 5.51 · 1017 cm−3 which can be easily achievable using the gas targets available at L2I [112] even with a low Z backing gas like helium. We have first performed simulations for this scenario. This virtual experiment simulates the interaction of the L2I laser beam focused with a 30 cm parabolic mirror onto a plasma of the required density. The vacuum Rayleigh range for this configuration is zR = 74.59 µm, and we will simulate propagation along ≈ 8zR . Table 6.2 shows the simulation parameters for this run. Simulation was done in 2D for 13000 timesteps, corresponding to a total laboratory time of 2.14 ps, or ∼ 640 µm of laser pulse propagation. The simulation took about 2 days to run using 32 cpus on the EP2 cluster. The laser pulse is focused onto the edge of a step like plasma density distribution and the simulation is done using a moving window. The simulation plasma is a cold plasma i.e. particles are injected into the simulation with zero velocity spread. The laser pulse is polarized linearly in the x3 direction (outside the simulation plane).

6.2 Laser Wakefield Accelerator ne 5.51 · 1017 cm−3 4.19 · 1013 rad/s ωp λ0 1.053 µm 5 µm w0 a0 3.23 215.6 × 146.4 µm2 sim. box sim. grid 4096 × 1024 cells 8 per cell num. particles Table 6.2: Simulation parameters for the laser wakefield accelerator run.

We have also included two low density test particle beams in the simulation with γ = 10 and γ = 50 respectively to study the acceleration of injected external beams. These beams have a uniform density along the x1 direction (the laser propagation direction) throughout all of the simulation volume and are injected in the center of the simulation box. Because they have a very low density (10−3 n0 ) they do not interfere with the background plasma, interacting only with the wakefield of the laser pulse.

6.2.1

Wakefield

Given the high a0 of the laser pulse the resulting ponderomotive force is extremely high. This results in the excitation of a strongly nonlinear plasma wave as it enters the plasma. However, after a short propagation distance, the wakefield structure evolves into a very clean and stable structure, mainly because the diffraction of the laser pulse lowers its intensity. Figure 6.3 shows the wakefield of the laser pulse at time t = 1.65 ps where the wakefield has stabilized in this regime. The plasma peak density at this time is 8.79 n0 (the plot is clipped at 6 n0 for a better display of the wave structure) and the peak accelerating field is 0.517 GV/cm in the leftmost accelerating bucket. This structure remains stable for the 350 µm of propagation simulated, with a slow decrease in the accelerating gradient. At the end of the simulation (t = 2.14ps) the laser pulse has diffracted significantly and the plasma peak density and peak accelerating field have decreased to 4.03 n0 and 0.399 GV/cm respectively. Throughout the simulation the peak values for these quantities are 34.47 n0 and 0.674 GV/cm. These values occur during the initial, strongly nonlinear stage, but only last for a short period of time, as we have mentioned above. We can also see in figure 6.3 that the wakefield wavefronts are curved and exhibit a typical horseshoe shape. This is related to the increase in plasma

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(a)

(b) Figure 6.3: Mass Density (a) and Accelerating Field (b) for the laser wakefield in the laser wakefield accelerator run.

6.2 Laser Wakefield Accelerator wavelength with increasing wave amplitude, as described in chapter 1. Since the plasma wave is driven more strongly on axis than it is off axis the plasma wavelength on axis is larger than off axis forcing the wavefronts to become curved [21, 113]. As we can see, the further back within the plasma wave train, the more curved the wave front, given that for this wavefront a greater dephasing between the on axis plasma wave and the off axis plasma wave has occured.

6.2.2

Particle Dynamics

The phasespace analysis (p1-x1) of this simulation shows particle trapping and acceleration for the background plasma. As we can see in figure 6.4, a significant part of the background plasma is trapped in the wakefield and accelerated to energies of up to 16.66 MeV. Since we are injecting a cold plasma into the simulation, background particles can only be trapped because of the wavebreaking that occurs in the initial stage of laser propagation and wakefield generation, which generates particles with high enough energy to become trapped in the laser wakefield. The same analysis applied to the test beams shows a more complicated behavior that can be seen in figure 6.4. The beam injected with γ = 10 (black on the plot) shows the typical orbits for particle trapping and acceleration in an accelerating wave structure. This beam reaches a maximum energy of 26.68MeV which corresponds to an average accelerating field of 0.416 GV/cm throughout the total simulation length. It is important to note that although the beam velocity is lower than the laser group velocity / wakefield velocity (eq. (1.12) gives γ ' 30 for these parameters), particle trapping still occurs which means that the particle trapping condition is not very stringent. Some of the beam particles are also decelerated down to energies of about 500keV as they interact with the decelerating portions of the wakefield. The beam injected with γ = 50 (red on the plot) is faster than the wakefield and no particle trapping occurs. These beam particles oscillate in the laser wakefield and get accelerated/decelerated accordingly. Still, energies of up to 54.1MeV are reached, which correspond to an average accelerating field of 0.446 GV/cm, slightly larger than for the γ = 10 beam, but still consistent with the above results. The decelerated particles have energies down to 1.82 MeV. The effect of the lower energy beam becoming trapped while the highest energy one only oscillating in the wakefield can be clearly seen in figure 6.6 where we show the 1D density along the propagation direction for the test beams (again the γ = 10 beam is represented in black and the γ = 50 beam

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Figure 6.4: Phasespace density (p1-x1) for the background electrons showing particle trapping and acceleration.

Figure 6.5: Phasespace plot (p1-x1) for the two test species: particles injected with γ = 10 (black) and particles injected with γ = 50 (red).

6.2 Laser Wakefield Accelerator

Figure 6.6: Electron density along the x1 direction for the two test species: particles injected with γ = 10 (black) and particles injected with γ = 50 (red).

is represented in red). The trapped beam (γ = 10) shows clear evidence of particle bunching, with a strong peak in density corresponding to the highest energies in each bunch, which corresponds to particles being trapped in the laser wakefield. For the higher energy beam however, no particle bunching is observed and very little variation in the density is observed along the propagation direction.

6.2.3

Energy Distribution

The final energy distribution for the two test beams, shown in figure 6.7, exhibits a similar saw tooth pattern for the two cases. For the γ = 10 beam the peaks on this spectrum closely match the maximum energy (and minimum energy for the peaks left to the injection energy) for each of the accelerated particle bunches. Note that figure 6.7 shows energy distribution in units of γ while figure 6.5 shows energy in units of MeV/c. For the γ = 50 beam a similar behavior is observed, with the peaks matching the maximum/minimum energy of beam particles on each cycle of the wakefield. Furthermore, we can see that a significant part of the γ = 50 beam is slowed down, as shown by the broadening of the γ = 50 peak, which is also related to the fact that this

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Figure 6.7: Energy density for the two test species, normalized to the maximum density: particles injected with γ = 10 (black) and particles injected with γ = 50 (red).

beam is not becoming trapped. This geometry shows excellent characteristics for a laser plasma accelerator but the wakefield can only be sustained for very short lengths because of laser pulse diffraction. Altough, energy gains of about 20 MeV can be accomplished in under 700 µm, for these accelerators to have practical use, the laser diffraction issue must be addressed. In fact, if we estimate the detuning length Ld using (1.13), we see that the acceleration only occurs for less than 1% of this length (Ld = 8.14 cm).

6.3

Channel Laser Wakefield Accelerator

To overcome the problem of the laser driver diffraction, tailored density plasma channels have been proposed to guide the laser pulses just like in a standard optical fiber. The L2I laboratory can now produce such channels using a novel laser triggered discharge system [28], and initial results indicate that this system can produce high quality, long length plasma channels for this effect. The optimal parameters for operating such channel laser plasma accelerators are still unknown and a detailed study is required. To that effect we have per-

6.3 Channel Laser Wakefield Accelerator

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λ0 1.053 µm 5 µm w0 a0 3.23 215.6 × 49.90 µm2 sim. box sim. grid 4096 × 512 8 per cell num particles Table 6.3: Simulation parameters common to all the channel electron accelerator runs.

τL /λp 0.5 1 2 3 5 10 20

ne [cm−3 ] 1.38 · 1017 5.51 · 1017 2.21 · 1018 4.96 · 1018 1.38 · 1019 5.51 · 1019 2.21 · 1020

ωp [rad/s] 2.09 · 1013 4.19 · 1013 8.38 · 1013 1.26 · 1014 2.09 · 1014 4.19 · 1014 8.38 · 1014

c/ωp [µm] ωL /ωp 14.31 85.41 7.157 42.71 3.579 21.35 2.386 14.24 1.431 8.541 0.716 4.271 0.358 2.135

1mm/Ld 1.53 · 10−3 1.22 · 10−3 9.78 · 10−2 3.29 · 10−1 1.53 1.22 · 101 1.04 · 102

Table 6.4: Simulation parameters used in each of the channel electron accelerator runs. The rightmost column shows the ration between the propagation length simulated and the corresponding dephasing length, Ld .

formed a parametric analysis of the optimal channel densities for this geometry in L2I laboratory conditions. Keeping the laser pulse length fixed at τL = 150 fs we simulated a laser pulse propagating in a matched parabolic channel for τL equal to 0.5λp , 1λp , 2λp , 3λp , 5λp , 10λp and 20λp , in the center of the channel. Tables 6.3 and 6.4 show the simulation parameters used for the different runs. The parabolic channel parameters were changed for each run to maintain a matched beam condition. According to equation (1.31), for a laser spot size of 10 µm in diameter, a channel depth nc = 4.51 · 1018 cm−3 is required for matched beam propagation. Furthermore, the channel characteristic width, r0 , must be equal to the laser spot size radius i.e. r0 = 5µm. In engineering units the plasma density profile will then be: n = n0 + nc

x2 = n0 + 1.8 · 1017 x2 [cm−3 ] r02

(6.1)

where x is the distance to the center of the channel in µm. In simulation units normalized to the plasma frequency in the bottom of the channel this density profile can be represented as:

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Figure 6.8: Channel density profile used in the simulations.

n 4 = 1 + 4 x02 n0 w0

(6.2)

where w0 is again the laser spot size radius in normalized c/ωp units and x0 the distance to the center of the channel normalized to the same units. Figure 6.8 shows a model of the transverse density profiles used. We decided on a total channel size (not to be confused with the characteristic channel size nc ) of three times the laser spot size diameter i.e. 30 µm. Also note that we are using leaky channels [35] that besides providing advantages in terms of wakefield stability and being less heavy to simulate using PIC code, more closely matches the channels produced at L2I. Along the propagation direction the channel has a step like density profile, beginning abruptly at the laser focal plane and remaining constant for the rest of the simulation space. We have also included a low density test beam in the simulation with γ = 50 in a similar way to what was done for the LWFA runs. This beam has an uniform density along the x1 direction (along the laser propagation direction) throughout all of the simulation volume and is injected in the center of the simulation box. Because of its very low density (10−3 n0 ) it does not interfere with the background plasma, interacting only with the wakefield of the laser pulse. Simulations were done in 2D for 21800 timesteps, corresponding to a

6.3 Channel Laser Wakefield Accelerator total laboratory time of 3.33 ps, and ∼ 1 mm of laser pulse propagation or ∼ 13.4 zR . The ratio between the total propagation length simulated and the corresponding dephasing lengths can be seen in table 6.4. Each simulation took about 1 day to run using 32 cpus on the EP2 cluster. The laser pulse is polarized linearly in the x3 direction (outside the simulation plane).

6.3.1

Laser Spot

The purpose of the plasma channel is to overcome laser diffraction. Figure 6.9 shows the laser spot size (radius) evolution along the propagation in the channel. The top plot presents these results in comparison to the laser spot size evolution in an uniform plasma, namely on the simulation performed earlier in this chapter, and the bottom plot shows a comparison between the multiple channel runs. Measurements were taken integrating the laser envelope along the propagation directions and fitting a gaussian function using a least squares method to the result. As we can see the laser pulse propagates through over 10 Rayleigh ranges with minimum variation of its spotsize as shown in comparison with the uniform plasma case. In fact, for the τL < 3λp runs, no significant increase in the average spot size is observable and for the τL = 3λp only a small growth in the average spot size is visible yet still in the order of 1 µm over 1 mm of propagation. Although we are using matched channels a small oscillation of the laser spot size can be observed in these results. This oscillation has a wavelength on the order of ∼ 200 µm that decreases as the plasma density increases for each run and appears to be an indication of a slight mismatch between the laser and channel parameters. This is caused by the high a0 of the laser pulse which further reduces the plasma density along the optical axis. This effect will increase the focusing characteristics of the channel causing the laser to over-focus and oscillate about an average spot size. Despite these oscillations, the laser pulse remains approximately gaussian both in the transverse and the longitudinal direction throughout the total length of the simulation. These results confirm the simulations presented in [11] that were done with a quasi-static code. For the τL ≥ 5 λp runs the laser pulse becomes self-modulated and spot size analysis becomes very difficult. Results are shown for the τL = 5 λp run only but as we can see after about 600 µm results become erratic. For the τL > 5 λp runs this effect happens much earlier and no valid results were obtained. However, the analysis of the laser envelope shows that the laser pulse remains within the channel for the propagating distance simulated.

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(a)

(b) Figure 6.9: Laser spot size evolution. (a) Shows a comparison between uniform plasma propagation and channel propagation and (b) compares the multiple channel regimes simulated.


6.3.2

Wakefield

The preformed plasma channel, by overcoming laser diffraction, allows for the accelerating wakefield to be sustained throughout long distances. The presence of this parabolic profile however has other effects on the laser plasma accelerator, namely changing the normal wakefield behavior, and requires detailed analysis. Figures 6.10 through 6.16 show the wakefield channel for all the simulation runs for a late time. The top plots show the mass density of the background plasma while the bottom plots show the associated accelerating gradient in the forward direction, x1. All results are presented for t ' 3 ps except for the τL = 10 λp and 20 λp runs where, because of the slower laser propagation velocity causing the wakefield to fall behind the simulation moving window, earlier times are presented. As we can see, the presence of the parabolic channel greatly affects the laser wakefield, and destroys it after a few (3-5) wakefield cycles. Given that the channel walls have a greater plasma density, the generated wakefield will have a much shorter wavelength in those regions than at the bottom of the channel, resulting in the destruction of the wakefield as the different wavelengths interact. Furthermore, the laser driver produces a shock wave that is reflected on the channel walls and interferes with the wakefield which also greatly affects its dynamics. In the τL ≥ 5 λp runs the laser pulse undergoes self-modulation and the wakefields produced are not as clean, becoming completely unstable for the higher density runs, and lasting only for short lengths. Under these conditions the accelerating structure is not stable enough to be of any use for a particle accelerator.

6.3.3

Time Evolution of the Accelerating Wake

A detailed analysis of the temporal evolution of the accelerating wake shows that despite the effects mentioned in the above section, the accelerating structure is sustained throughout the total simulation time for the τL ≤ 3 λp runs, demonstrating the feasibility of the channeled laser plasma accelerator for these conditions. Figures 6.17 to 6.23 show these results. The plots show the time evolution (horizontal axis) of the 1D accelerating gradient taken at the center of the channel (vertical axis). As we can see the τL ≤ 3 λp runs remain stable and show little evolution throughout the run. The small oscillations present in these plots are related to the oscillations in the laser spot size discussed in

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(a)

(b) Figure 6.10: Density Wake (a) and Accelerating Field (b) for the τL = 0.5λp run.


(a)

(b) Figure 6.11: Density Wake (a) and Accelerating Field (b) for the τL = 1λp run.

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136


(a)



(a)


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(a)



(a)


139

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(a)



Figure 6.17: Time evolution of the accelerating wake for the τL = 0.5λp run.

a previous section. Furthermore, the wakefield appears to become more stable as the run progresses which indicates that longer channels will be efficient in maintaining the wakefield for longer propagation lengths. The τL > 5 λp runs, although reaching much higher accelerating fields, have a limited duration, and get destroyed after a short propagation because of laser pulse self modulation, rendering the use of a plasma channel useless. It is also clear that these wakefields have a much lower phase velocity. It should be noted that the simulation window is moving with the speed of light, so for these simulations the laser pulse and the wakefield fall significantly behind the numerical box. However, for the time where the wakefield is present, the region of interest falls well inside the simulation window, thus validating these results. The τL = 5 λp run is in the transition between the normal LWFA regime and the SMLWFA regime and represents a special case. It takes about 400 µm of laser propagation before we get an intense accelerating gradient, which is virtually a single cycle wakefield. An interesting effect occurs after about 600 µm where a bunch of accelerated electrons gets ejected from the plasma background. This will be discussed in detail further ahead in this chapter.

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Figure 6.18: Time evolution of the accelerating wake for the τL = 1λp run.





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Figure 6.24: Wakefield phase velocity (γp ) for all the regimes simulated. The measured velocities are marked as circles and the solid lines show several theoretical models: (a) is the 1D limit, (b) includes the 2D (curvature) corrections, (c) is the 1D limit using the corrected density and (d) is the same as (c) using the corrected density.

6.3.4

Wakefield Phase Velocity

The results presented in the previous section allow us to determine another key parameter for laser plasma accelerators, namely the wakefield phase velocity. This is especially important as it determines the dephasing length and the maximum energy that can be gained in the laser plasma accelerator as described on chapter 1. Figure 6.24 shows the wakefield phase velocity for the different simulations. Results shown are average results, as this phase velocity varies slightly along the run. The figure also shows results from several theoretical models for comparison. The observed wakefield phase velocity deviates significantly from the 1D limit, γp ' ωL /ωp as calculated using the density at the bottom of the channel, and assuming that the wakefield phase velocity will be the same as the laser group velocity. Even including curvature corrections as described by (1.12) the simulation results are still much lower than the theoretical ones. This is a result from the fact that a significant portion of the laser pulse is propagating trough a region with higher plasma density, which therefore has a lower group

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Simulation of Laser Acceleration Experiments at L2I velocity. This indicates that we should use an effective density to account √ for the channel effects. If we consider an effective density of nef f = n0 + 2nc we get reasonable agreement for the τL < 5 λp runs. For the τL ≥ 5 λp runs the self modulation of the laser pulse results in an even lower wakefield phase velocity. These results imply that the higher density configurations, τL ≥ 5 λp , will have a much shorter dephasing length, which will result in lower maximum energies despite the stronger accelerating gradient predicted by (1.8). For a laser plasma accelerator to be feasible at these higher densities we would require a much shorter wavelength than the one available at L2I, so that the laser group velocity (and hence the wakefield phase velocity) would be much higher. One interesting aspect would be to test these concepts using a frequency doubled beam. Although this would mean a lower a0 the increase in laser group velocity might prove to be advantageous. Furthermore, it is not at all obvious that higher a0 means a better accelerating structure.

6.3.5

Wakefield wavelength

The wakefield wavelength is of relevance to determine the injection region and the size of the accelerated bunches. The fact that the wakefield exists in a region of parabolic plasma density also strongly affects the wakefield wavelength. The simulations results are significantly different from the expected value of λp as calculated at the bottom of the channel. Figure 6.25 shows the simulation results for the wakefield wavelength in comparison with several theoretical models. For the lower densities, τL ≤ 3, we have nc n0 and the channel walls cause the effective wavelength to decrease significantly. The effective density determined in the previous section however appears to be a too strong correction, and a smaller density would give better results. For the higher densities the nonlinear effect of wavelength increase dominates and the wavelength is larger than λp as predicted by equation (1.11).

6.3.6

Maximum Accelerating Gradient

The maximum accelerating gradient, and the dephasing length yield the maximum energy a laser plasma accelerator can reach. In a channel accelerator this is especially important as the wakefield can be sustained for longer propagation lengths. However, it should be noted that higher accelerating fields do


Figure 6.25: Wakefield wavelength in the bottom of the channel (λp0 ) for all the regimes simulated. The measured wavelengths are marked as circles and the solid lines show several theoretical models: (a) is the 1D limit, (b) is the 1D limit using the corrected density as determined for the wakefield phase velocity and (c) is the same as (b) using the nonlinear correction.

not mean a better accelerator by themselves; the accelerating structure must remain coherent with the particles being accelerated for as long as possible. Figure 6.26 shows the peak accelerating gradient for the simulations run in comparison with theoretical models. As we can see, for the low density τL < 3 λp runs the measured peak accelerating gradient lies between the 1D non-relativistic and relativistic wavebreaking limits calculated at the bottom of the channel. For higher densities the peak accelerating field increases and has a peak value for the τL = 10 λp run. The extremely high accelerating gradient exceeds all theoretical models, even using the effective density mentioned in previous sections and corrected wakefield phase velocities. However, as mentioned previously, this accelerating structure lasts only for a short propagation length, therefore not being especially useful in a plasma accelerator. Finally, for the τL = 20 λp run, the peak accelerating gradient decreases again, as the strong self modulation of the laser does not allow the wakefield to grow.

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Figure 6.26: Maximum accelerating gradient for all the regimes simulated. The measured accelerating gradients are marked as circles and the solid lines show several theoretical models: (a) is the nonrelativistic wavebreaking limit, (b) is the relativistic wavebreaking limit, (c) is the same as (a) using the corrected density and (d) is the same as (b) using the corrected density and the corrected γp .

6.3.7

Particle energy

Having analyzed the accelerating structure we now look at the evolution of particle energies, namely the background plasma energy and the injected beam. Figure 6.27 shows the maximum energy for the background plasma and the injected beam as a function of laser propagation. Since we are considering a cold plasma, the lower density runs with τL ≤ λp show little evidence of background plasma particle trapping and acceleration. For larger densities, some background particles get trapped and gain energies of up to 200MeV in the τL = 5λp , experiencing a near constant accelerating gradient. It should be noted however that very few background particles do get trapped in these simulations with the exception of the τL = 5λp run, as we shall see in the next section. For larger densities, i.e. τL > 5λp , we observe a clear saturation of the maximum energy, as predicted by the wakefield analysis done above. Regarding the injected beam, a peak energy of 280.93 MeV is reached

6.3 Channel Laser Wakefield Accelerator for the τL = 3λp run. For the lower density runs, with τL ≤ 3λp we observe a steady energy gain of the beam particles which increases with density, as it would be expected. The average accelerating gradient is 1.04, 1.22, 1.88 and 1.97 GV/cm for the τL = 0.5, 1, 2 and 3 λp runs respectively. The best cases are the τL = 2 and 3λp cases, as they present the largest energy gain, and show the most stable energy evolution. For the τL = 5λp a strong energy gain is still observed, but the behavior is a little more erratic. Two different stages can be identified for this run: the first until ∼ 700 µm of propagation, where the maximum energy is obtained in the region where a small amplitude wake, visible in figure 6.21, is present. As these particles outrun the plasma wave and start to lose energy, those in the stronger wake region closer to the laser driver become the ones with the highest energy and then evolve smoothly until the end of the run. For larger densities, i.e., τL > 5λp we observe a clear saturation of the maximum energy, just like in the background plasma analysis. It should also be noted that we are far from the dephasing limit predicted by standard wakefield theory, as shown in table 6.4, so to fully understand the limitations of these accelerators simulations of longer propagation lengths will be required. Given the average accelerating gradient of 1.88 GeV/cm observed, using such an accelerator with τL ∼ 2λp should allow for energy gains of up to 10 GeV in only a few centimeters. Also note that, as mentioned above, the use of a shorter wavelength laser driver can improve on the energy limitations of this accelerator. Figure 6.28 shows the energy spectra for the τL = 2λp run. All runs with τL ≤ 3λp present similar results. The energy distribution of the background plasma shows a thermal distribution with a power law tail of energetic electrons, that is compatible with experimental observations of the background energy of the electrons for self-modulated laser wakefield experiments (see for example [52]). The injected beam shows a more complicated structure, with a saw-tooth pattern similar to the one observed in the LWFA simulations, where each peak corresponds to an accelerated bucket. We also observe a deceleration of a significant part of the injected beam, which can be seen as a widening of the γ = 50 peak. A phasespace analysis (p1-x1) was also done but no extra information is obtained from it. Regarding the accelerated beams divergence, by analyzing the p2-p1 phasespace we observe that the accelerated beam always show divergences below 5 degrees.

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(a)

(b) Figure 6.27: Evolution of the maximum energy of the background plasma (a) and the injected particles (b).


(a)

(b) Figure 6.28: Energy distribution for the τL = 2λp run, for t ' 3.3ps for the background plasma (a) and the injected particles (b).

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6.3.8

Ejected bunch

One important issue with laser plasma accelerators is that of particle injection for acceleration. As we have seen, the background plasma does not produce a significant amount of accelerated particles, and an injected beam is required to generate high energy electrons in sufficient quantity. An interesting effect occurs for the τL = 5λp that is of special relevance for this issue. After ∼ 580 µm of laser pulse propagation a high charge, high energy electron bunch gets ejected from the plasma background creating a well defined electron pulse. As the laser propagates through the plasma the self-focusing effect increases its local intensity and creates a long region where virtually no background plasma exists. The result is a very intense electric field that takes particles from the background plasma and accelerates them to high energies. Since for this run the laser has a small group velocity, these particle soon enter into the region without plasma, thus creating a clean ejected bunch. At the final stage of this simulation the electron bunch is overtaking the laser pulse and is oscillating transversely in its electromagnetic field. Figure 6.29 shows this ejected bunch. The bunch as a length of 5.1µm or 17fs, a peak density for this bunch is 22.8 times the background density, and has a total charge of this bunch is about 5.4×109 electrons or ∼ 0.86 nC. The total charge for this bunch is of the same order of magnitude as the one observed in a related work [114] that has recently been presented. In that work, a 33 fs, 12 J, with τL = 0.5λp laser pulse focused onto an uniform plasma, would produce a ∼ 5 nC electron bunch. Our results show the possibility of generating such electron bunches using today’s laser technology. It should be noted however that the electron bunch is close to overtaking the laser pulse completely and that no further acceleration should be expected after that point. However, this bunch is ideal to inject in another accelerating structure. Figure 6.30 shows a section of the p1-x1 phasespace showing the momentum of the ejected bunch. This bunch has energies of up to 100 MeV and is clearly separated from the remaining background plasma. In this plot we can also clearly see a second bunch trailing the first one with lower density but with energies going up to 200 MeV, that is also visible in figure 6.29. Another important result is the emittance of the produced electron bunch. Figure 6.31 shows the p2-p1 phasespace for this run at the same time as the previous plots. The ejected bunch has a divergence below 5 degrees, and remains constant throughout the simulation after the bunch has been ejected. Since the peak accelerating gradient for this simulation is much higher than for the τL < 5λp runs we may expect to see this effect on the lower


Figure 6.29: Section of the mass density for the τL = 5λp run, for t ' 3.3 ps showing the ejected particle bunch (dotted circle).

Figure 6.30: Section of the p1x1 phasespace density for the τL = 5λp run, for t ' 3.3 ps.

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Figure 6.31: Plot of the p2p1 phasespace density for the τL = 5λp run, for t ' 3.3ps showing the ejected particle bunch (dotted circle).

density scenarios for a later time. In fact, it should be noted the the ejected bunch appears around the time the laser as propagated for one dephasing length which for this simulation is ∼ 650 µm so for the same effect to occur in the τL = 2λp condition we may have to propagate the laser pulse for over 10 mm. However, it should also be noted that for the lower density conditions the laser pulse will not be self modulated and no increase in the peak intensity will occur, and that these conditions are important for the ejected beam to appear.

6.4

Conclusions

We have presented in this chapter the numerical modeling of laser acceleration experiments under the L2I laser laboratory conditions. Given the high intensities involved in these situations fluid and reduced codes cannot capture the fundamental issues involved and PIC code simulations are essential to accurately model these experiments. In this sense, the use of the OSIRIS framework was of capital importance, providing the necessary toolkit to establish the feasibility of experimental parameters for the L2I laboratory. These results show that the prospects for an experimental demonstra-

6.4 Conclusions tion of laser-plasma accelerator at L2I are extremely promising. Operating under the LWFA conditions a very clean accelerating wakefield can be produced that can be used for the acceleration of injected beams. Although limited by laser diffraction, the simplicity of this setup is its main advantage, and as we have seen high energy gains can be obtained in only tens of microns. The parametric analysis of the channeled laser plasma accelerator showed that the best results in this regime are obtained for τL ' 2, 3 λp , providing excellent acceleration for injected beams. This would correspond to a plasma density at the bottom of the channel of ne ∼ 3 · 1018 cm−3 which is well within the operating parameters of the L2I channel setup. Although we have only simulated propagation for a 1 mm channel, the results look very promising for longer propagation distances, which would result in much larger energy gains. In these situations we should also investigate if lower laser intensities would produce more stable wakefields, as the lower accelerating gradients could be compensated by the longer propagation lengths. We have also shown that neither the LWFA nor the channeled laser plasma accelerator can serve as particle sources on their own, given that for the most cases no proper accelerated electron beam is produced from the plasma background. In this sense the novel regime identified in the τL = 5 λp runs is of special importance. The well defined, high energy electron bunch produced can be used as a 1 mm long 100 MeV electron source with multiple applications. Further work on this regime is required to properly identify the optimal parameters for this injection mechanism but current results look very promising.

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Chapter 7 Overview and Future Work The field of laser-plasma accelerators has grown significantly over the past few years, with the advent of new laser technology and new accelerating geometries being proposed frequently. In this thesis we presented the numerical and experimental work on this field developed over the past five years at the Grupo de Lasers e Plasmas in the Instituto Superior Técnico of the Universidade Técnica de Lisboa, Portugal. We described the development and implementation of a magnetic electron spectrometer in the 10 − 200 MeV range for use in laser plasma acceleration experiments, and the numerical modeling of proposed experiments at the L2I laboratory, with special emphasis on the channel laser-plasma accelerator. We also described the development and implementation of the EP2 cluster for numerical computing where all the simulation work was done. This work fitted the L2I laboratory with a fundamental diagnostic tool for laser plasma accelerator studies that will allow in the future the detailed characterization of accelerated particle spectra. We have successfully identified the optimal parameters for channel laser plasma accelerator experiments, and a novel regime for particle injection from the background plasma. We would like to acknowledge the collaboration with the Rutherford Appleton Laboratory, in the UK, and the UCLA Plasma Simulation Group, in the USA. The electron spectrometer was designed specifically with laser-plasma accelerator experiments in mind and has a number of characteristics specially defined for this purpose. Although based on a time proved design, the magnetic system includes two significant alterations that allow for greater vertical focusing and resulting brightness, and for the use of this system as an integrating spectrometer for higher energy analysis. The operating energy range in the spectrograph configuration is especially suited for laser-plasma accelerator experiments and the entire spectrum can be swept in only three shots. This

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Overview and Future Work spectrometer accepts high divergence beams maintaining high accuracy and, compared with other systems of this type, is relatively light and portable given its custom design. The detection system used for this spectrometer is based on scintillation techniques that convert particle kinetic energy into light. Detection in this range of energies presents some dificulties as commercial detectors available are extremely expensive. The development of our own detection system resulted in a more cost effective solution and also allows for greater versatility and understanding of the fundamental behavior of such system. The use of two complementary systems achieves a good balance between energy and particle count accuracy and the discrete channel system is easily expandable to a large array of detectors. Furthermore, although the L2I laser system has a repetition rate of about one shot every two minutes, higher repetition rate laser systems can be used with these detectors, making it usable in several laser laboratories worldwide. Another upgrade being planned is the use of a scintillating fiber array connected to a CCD camera. This may provide a lower cost solution for adding more channels to the discrete system if we disregard the price of the CCD camera that already exists at the L2I laboratory for other purposes. The mechanical integration of the spectrometer system presented some special difficulties given the weight and precision requirements. The custom stand built for this purpose can be easily adapted for use with other experimental devices like a secondary vacuum chamber, and allows for great stability and alignment precision. The custom designed vacuum chamber allows for operation at low energies, setting the lower limit of the operation to 1 MeV, and allowing for greater accuracy in the measurements. Furthermore, the techniques developed for signal acquisition are applicable to any charge producing detectors (e.g. photodiode based detectors) where only the integrated/average signals are of interest, and are applicable to high repetition rate experiments. Given the high intensities involved in laser-plasma accelerator experiments fluid and reduced codes cannot capture the fundamental issues involved and PIC code simulations are essential to accurately model these experiments. We have presented the OSIRIS framework for modeling plasma based accelerators. It should be noted that this is an ongoing effort in a collaboration involving over several researchers over four institutions; future developments will concentrate on the implementation of true open-space boundaries, ionization routines and dynamic load balancing. Regarding the visualization and data analysis infrastructure, a Web-Driven visualization portal will be implemented on the near future, allowing for efficient remote data analysis on clusters. The installation of the EP2 cluster was fundamental for the simulation

159 work presented in this thesis, allowing for very fast turnaround times and flexibility on the modeling of the laser-plasma accelerator experiments at the L2I laboratory. To our knowledge, this the first Macintosh G4 cluster in Europe, and is currently the highest performance parallel computing machine in Portugal. The acquired know-how on cluster computing is fundamental for future developments in laser-plasma accelerator modeling at our group. The planned upgrades in terms of the number of CPU’s and network backbone will further improve our computational capabilities allowing for bigger simulations, and faster turnaround times. The goal of one to one modeling of the laser-plasma accelerator experiments at the L2I laboratory in 2D was successfully achieved using the OSIRIS framework on the EP2 cluster. The main results were the characterization of the laser wakefield accelerator regime for the current operational parameters of the L2I laser system and the parametric analysis of the channel laser plasma accelerator. We have shown the feasibility of laser wakefield accelerator experiments at the L2I laboratory achieving accelerating gradients of about 500 MeV/cm. Regarding the channel laser-plasma accelerator we have identified the optimal operating regime as having a laser pulse duration of about two times the plasma collisionless skin depth at the bottom of the channel, sustaining accelerating gradients of almost 2 GeV/cm for the propagation distance simulated. We have discarded the use of longer pulses given that the laser guiding is assured by the channel and that the self modulation instability simply destroys the laser pulse driver. A more detailed analysis of these scenarios, including propagation for longer lengths, measured channel profiles and different intensies will complete this study but we fill the main parameters have been established. We have also identified a novel regime identified in which a well defined, high energy electron bunch is produced from the background plasma that can be used as a 1 mm long 100 MeV electron source with multiple applications. As a final remark, it should be noted that present and future experiments are still far from the parameter range of interest to high energy physics. The energy range, total charge and beam luminosity are still a few of orders of magnitude below conventional accelerator technology, and far from producing acceptable event rates. However, this is an emerging technology that has shown great progress in the recent years and the work presented in this thesis shows the feasibility of producing good beam quality 100 MeV - 1 GeV electrons with present technology, which is a major milestone in laser-plasma accelerator technology.

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