University of California Los Angeles Nonlinear Plasma Wakefield Theory and Optimum Scaling for Laser Wakefield Accelerator (LWFA) in the Blowout Regime A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Electrical Engineering by Wei Lu 2006 (cid:13)c Copyright by Wei Lu 2006 The dissertation of Wei Lu is approved. Steve Cowley F.F. Chen Chan. Joshi Warren B. Mori, Committee Chair University of California, Los Angeles 2006 ii Chasing the light ...... To my family for their support, understanding, and above all, love ...... iii Table of Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Plasma Based Acceleration . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Particle-in-Cell Simulation . . . . . . . . . . . . . . . . . . . . . . 11 1.4 Motivation and Outline . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Nonlinear Theoryfor RelativisticPlasmaWakefieldsinthe Blowout Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 The Basic Equations for Wake Excitation . . . . . . . . . . . . . . 20 2.3 On Blowout and Sheet Crossing . . . . . . . . . . . . . . . . . . . 26 2.4 Wake Excitation in the Blowout Regime . . . . . . . . . . . . . . 33 2.5 On the Transition between Linear Theory and the Breakdown of Fluid Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6 The Ultra-relativistic Blowout Regime . . . . . . . . . . . . . . . 45 2.7 Formulas for Arbitrary Blowout Radius . . . . . . . . . . . . . . . 47 2.8 Differences between Laser Driver and Electron Beam Driver . . . 49 2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3 Some Applications of the Nonlinear Wakefield Theory . . . . . 54 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 iv 3.2 The optimum Plasma Density for Plasma Wakefield Excitation in the Blowout Regime . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Beam Loading in the Blowout Regime . . . . . . . . . . . . . . . 61 3.4 The Transformer Ratio for a Ramped Electron Beam . . . . . . . 65 3.5 The Electron Hosing Instability in the Blowout Regime . . . . . . 68 3.5.1 What is the Electron Hosing Instability? . . . . . . . . . . 68 3.5.2 Toward a more General Hosing Theory for a Narrow Elec- tron Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4 On Wave Breaking and Particle Trapping in Plasma Waves . . 81 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.1.1 What Does “Wave Breaking ” Mean in One-dimensional Plasma Waves . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.1.2 What Does “Wave Breaking ” Mean in Multi-dimensional Plasma Waves . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.1.3 “Periodic ” vs. “Driven” Waves . . . . . . . . . . . . . . . 86 4.2 GeneralFormalismforParticleMotioninFieldswithTranslational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.2.1 The General Trapping Condition . . . . . . . . . . . . . . 93 4.2.2 The General Particle Energy Bounds . . . . . . . . . . . . 94 4.3 Wave Breaking of Driven Plasma Waves . . . . . . . . . . . . . . 97 5 LWFA Scaling in the Blowout Regime . . . . . . . . . . . . . . . 103 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 What Do We Need for a LWFA as an Useful Accelerator? . . . . . 106 v 5.3 What Does Linear Theory Tell Us? . . . . . . . . . . . . . . . . . 107 5.3.1 Stability Consideration . . . . . . . . . . . . . . . . . . . . 108 5.3.2 Efficiency Consideration: Pump to Wake . . . . . . . . . . 110 5.3.3 Laser Guiding . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.3.4 Beam Loading and Beam Quality . . . . . . . . . . . . . . 114 5.3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.4 Does Nature Force Us to the Blowout Regime? . . . . . . . . . . . 118 5.5 Physics in the Blowout Regime . . . . . . . . . . . . . . . . . . . 122 5.5.1 Blowout and Matching Condition . . . . . . . . . . . . . . 123 5.5.2 Wake Excitation . . . . . . . . . . . . . . . . . . . . . . . 126 5.5.3 Local Pump Depletion and Laser Front Etching . . . . . . 128 5.5.4 Guiding : Self-Guiding or Channel Guiding . . . . . . . . . 133 5.5.5 Injection and Beam Loading . . . . . . . . . . . . . . . . 136 5.5.6 Possible Laser Plasma Instabilities and Laser Pulse Distor- tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.6 Phenomenological Scaling and Its Verification by PIC Simulations 140 5.7 Comparison with the Scaling Based on Similarity Theory . . . . 148 5.8 Parameter Design for Future . . . . . . . . . . . . . . . . . . . . . 152 6 Prospects for Plasma Based Acceleration in the Blowout Regime159 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 vi List of Figures 2.1 Electron charge density with the defined blowout radius r (ξ) . . . 17 b 2.2 Plots of trajectories for electrons at different initial radial position √ for an electron beam driver with k σ = 0.01 , k σ = 2, the p r p z beam center ξ = 5 and with (a) n = 1 (b) n = 10 . . . . . . . 29 0 b0 b0 2.3 J /c−ρ profile from a PIC simulation . . . . . . . . . . . . . . . 36 z 2.4 Comparison of the trajectories of r (ξ) and the accelerating field b E (ξ) between theoretical calculations and PIC simulations: PIC z simulation(red), calculation using a constant profile (green), and calculation using a varying profile (blue). The maximum blowout radius is (a) r = 4 (b) r = 4 with two beams (c) r = 2 (d) m m m r = 0.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 m 2.5 Plasma density plots from PIC simulations with matched and un- matched spot sizes for the same laser power P and plasma density n , P/P = 8 (a) matched case with k W = 4 and a = 4 (b) p c p 0 0 unmatched case with k W = 3 and a = 5.3 . . . . . . . . . . . . 52 p 0 0 3.1 The normalized beam dimensions k σ , k σ and the normalized p r p z blowout radius k r (the driver is a bi-Gaussian beam with N = p m 1.8×1019, σ = 32µm and σ = 10µm) . . . . . . . . . . . . . . . 58 z r 3.2 Thenormalizedandabsolutepeakwakefieldamplitudes(thedriver is a bi-Gaussian beam with N = 1.8 × 1019, σ = 32µm and z σ = 10µm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 r vii 3.3 Plasma wake driven by an electron beam with a linearly ramped current profile n /n = 100, k a = 0.5, L = 22 (a) plasma phase b p p 0 space x2x1 (b) color plot of E1 (c) lineout of E1 . . . . . . . . . . 78 3.4 A plot of the beam on top of the plot of the ion channel in the nominal “afterburner” simulation. . . . . . . . . . . . . . . . . . . 79 3.5 The centroid oscillation |x | (blue curve) in a self-generated chan- b nel and the prediction (red curve) from the fluid theory for an equilibrium channel. The black line is a linear fit for the initial growth in the simulation before the nonlinearity occurs. This ini- tial growth is one order of magnitude smaller than the result for a equilibrium channel. . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.6 Density plots of the beams (red) and plasma (blue) in (a) the adia- batic non-relativistic regime; (b) the adiabatic relativistic regime; (c)thenon-adiabaticnon-relativisticregime; (d)thenon-adiabatic relativistic regime. The beams move to the left in these plots. . . 80 3.7 Hosing growth in four regimes. . . . . . . . . . . . . . . . . . . . 80 4.1 A weakly nonlinear coherent wake field driven by an 1D electron driver: n /n = 0.1, k σ = 1.5 (a) color plot of E1 (b) lineout of b p p z E1 (c) lineout of ψ (d) phase space p1x1 . . . . . . . . . . . . . . 99 4.2 A strongly nonlinear wake field driven by an 1D electron driver: n /n = 0.1, k σ = 1.5 (a) color plot of E1 (b) lineout of E1 (c) b p p z lineout of ψ (d) phase space p1x1 (e) ψ near −1 . . . . . . . . . . 100 viii 4.3 A strongly nonlinear wake field driven by an 3D electron driver: √ n /n = 100, k σ = 2, k σ = 0.5 (a) color plot of phase space b p p z p r x2x1 (b) color plot of E1(c) lineout of E1 (d) lineout of ψ (e) ψ near −1 (f) phase space p1x1 . . . . . . . . . . . . . . . . . . . . 101 5.1 A laser pulse with a = 1, k cτ = π propagating through a plasma 0 p withn /n = 0.00287,(a)E3attimet = 400ω−1 (b)E3attimet = p c 0 20400ω−1 (c) E1 at time t = 400ω−1(d) E1 at time t = 20400ω−1 111 0 0 0 5.2 A laser pulse with a = 2, k cτ = π propagating through a plasma 0 p withn /n = 0.00287,(a)E3attimet = 800ω−1 (b)E3attimet = p c 0 20400ω−1 (c) E1 at time t = 800ω−1(d) E1 at time t = 20400ω−1 112 0 0 0 5.3 A 30fs, 6TW short laser pulse with a = 2, k cτ = π and k W ≈ 0 p p 0 2.83 propagating through a plasma channel with n /n = 0.00287 p c on axis, note that k cτ /π = 0.7 and k cτ /π = 1.3 . . . . . . 121 p rise p fall 5.4 Plasmadensityplotsformatchedblowoutofalaserdriver, P/P = c √ 1, a = 2, k W = 2 2, k cτ = π and n /n = 0.00287, (a) at 0 p 0 p p c 0.57Z (b) at 2.85Z (c) at 6.27Z (d) at 10Z . . . . . . . . . . . 127 r r r r 5.5 Lineouts of E3 for 1D and 3D simulations for a = 2 andn /n = 0 p c 0.00287 at t = 2500ω−1 (a) 1D simulation (b) 3D simulation with 0 √ laser spot size k W = 2 2 and a plasma channel depth ∆n /n = p 0 c p 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.6 Lineouts of E3 for 1D and 3D simulations for a = 40 andn /n = 0 p c 0.04 at t = 12000ω−1 (a) 1D simulation (b) 3D simulation with 0 matched laser spot size k W = 10 . . . . . . . . . . . . . . . . . . 131 p 0 ix