Lecture 3: Laser Wake Field Acceleration (LWFA) 1D-Analytics: Nonlinear Plasma Waves 1D Wave Breaking Wake Field Acceleration Bubble Regime (lecture 4): 3D Wave Breaking and Self-Trapping Bubble Movie (3D PIC) Experimental Observation Bubble Fields Scaling Relations

Direct Laser Acceleration versus Wakefield Acceleration Pukhov, MtV, Sheng, Phys. Plas. 6, 2847 (1999) plasma channel E B laser electron Free Electron Laser (FEL) physics DLA acceleration by transverse laser field Non-linear plasma wave LWFA Tajima, Dawson, PRL43, 267 (1979) acceleration by longitudinal wakefield

Laser pulse excites plasma wave of length lp= c/wp -0.2 0.2 eEz/wpmc 2 -2 eEx/w0mc -20 20 px/mc 40 g Z / l 270 280 3 -3 zoom -0.2 0.2 eEz/wpmc wakefield breaks after few oscillations 40 20 g What drives electrons to g ~ 40 in zone behind wavebreaking? lp laser pulse length Laser amplitude a0 = 3 Transverse momentum p/mc >> 3 p /mc zoom 3 -3 a 20 -20 Z / l 270 280 l z

How do the electrons gain energy? -2x103 0 103 0 2x103 G|| G dt p2/2 = e E  p = e E|| p|| + e E p dt p = e E + v  B e c G = 2 e E pdt   Gain due to transverse (laser) field: G|| G 0 104 0 104 G|| = 2 e E|| p|| dt   Gain due to longitudinal (plasma) field:

Phase velocity and gph of Laser Wakefield lp density laser Short laser pulse ( ) excites plasma wave with large amplitude. Light in plasma (linear approximation)

1D Relativistic Plasma Equations (without laser) Consider an electron plasma with density N(x,t), velocity u(x,t), and electric field E(x,t), all depending on one spatial coordinate x and time t. Ions with density N0 are modelled as a uniform, immobile, neutralizing background. This plasma is described by the 1D equations: cold plasma

Problem: Linear plasma waves Consider a uniform plasma with small density perturbation N(x,t)=N0+N1(x,t), producing velocity and electric field perturbations u1(x,t) and E1(x,t) ,respectively. Look for a propagating wave solution Show that the 1D plasma equations, keeping only terms linear in the perturbed quantities, have the form giving the dispersion relation Apparently, plasma waves oscillate with plasma frequency for any k, in this lowest order approximation, and have phase velocity vph=wp/k. Show that for plasma waves driven by a laser pulse at its group velocity ( ), one has

10. Problem: Normalized non-linear 1D plasma equations show that the the 1D plasma equations reduce to We now look for full non-linear propagating wave solutions of the form Using the dimensionless quantities

Nonlinear 1D Relativistic Plasma Wave 1. integral: energy conservation

Wave Breaking density spikes diverge t Maximum E-field at wave breaking (Achiezer and Polovin, 1956) Non-relativistic limit (Dawson 1959)

11. Problem: Derive non-linear wave shapes Show that the non-linear velocity can be obtained analytically in non-relativistic approximation from with the implicit solution Notice that this reproduces the linear plasma wave for small wave amplitude bm. Then discuss the non-linear shapes qualitatively: Verify that the extrema of b(t), n(t), and the zeros of E(t) do not shift in t when increasing bm, while the zeros of b(t), n(t), and the extrema of E(t) are shifted such that velocity and density develop sharp crests, while the E-field acquires a sawtooth shape.

Wakefield amplitude The wake amplitude is given between laser ponderomotive and electrostatic force Using with for circular polarization, one finds density laser For linear polarization, replace .

Time between injection Dephasing length Acceleration phase E-field t lp Emax Dt Time between injection and dephasing Dephasing length Estimate of maximum particle energy

FLUID VS. TRAPPED ORBITS Viewgraph taken from E. Esarey Talk at Dream Beam Symposium www.map.uni-muenchen.de/events.en.html UID: symposium PWD: dream beams PHASE-SPACE ANALYSIS FLUID VS. TRAPPED ORBITS trapped orbit (e- “kicked” from fluid orbit) 1D case: Trapped electrons require a sufficiently high momentum to reside inside 1D separatrix 1D separatrix cold fluid orbit (e- initially at rest)

Maximum electron energy gain Wmax in wakefield Electron acceleration (norm. quantities) acceleration range For maximum wave amplitude (in units, first obtained by Esarey, Piloff 1995)

Wave Breaking single electron motion injected at phase velocity (bg)ph p/mc = bg collective motion of plasma electrons single electron motion injected at phase velocity E/E0 Longitudinal E-field Wave-Breaking at p/mc = b

Example Plasma: Laser: E-field at wave-breaking: Dephasing length: Required laser power:

Plasma filled capillary Nature Physics 2, 456 (2006) L=3.3 cm, f=312 mm Laser 1.5 J, 38 TW, 40 fs, a = 1.5 Plasma filled capillary Density: 4x1018/cm3 Divergence(rms): 2.0 mrad Energy spread (rms): 2.5% Charge: > 30.0 pC 1 GeV electrons

GeV: channeling over cm-scale Increasing beam energy requires increased dephasing length and power: Scalings indicate cm-scale channel at ~ 1018 cm-3 and ~50 TW laser for GeV Laser heated plasma channel formation is inefficient at low density Use capillary plasma channels for cm-scale, low density plasma channels Capillary Plasma channel technology: Capillary 1 GeV e- beam Laser: 40-100 TW, 40 fs 10 Hz 3 cm

225 mm diameter and 33 mm length capillary 0.5 GeV Beam Generation 225 mm diameter and 33 mm length capillary Density: 3.2-3.8x1018/cm3 Laser: 950(15%) mJ/pulse (compression scan) Injection threshold: a0 ~ 0.65 (~9TW, 105fs) Less injection at higher power -Relativistic effects -Self modulation a0 Stable operation 500 MeV Mono-energetic beams: a0 ~ 0.75 (11 TW, 75 fs) With 225 micron diameter and 33 mm length capillary, stable generation of half GeV mono-energetic beams was found. This is a typical image from the spectrometer. Horizontal axis is Energy in MeV, and vertical axis is space. Input laser parameter was about 11 TW 75 fs. For this particular shot, peak energy was 490 MeV, divergence was 1.6 mrad, energy spread was 5.6 %, and charge was around 50 pC. Peak energy: 490 MeV Divergence(rms): 1.6 mrad Energy spread (rms): 5.6% Resolution: 1.1% Charge: ~50 pC

312 mm diameter and 33 mm length capillary 1.0 GeV Beam Generation 312 mm diameter and 33 mm length capillary Laser: 1500(15%) mJ/pulse Density: 4x1018/cm3 Injection threshold: a0 ~ 1.35 (~35TW, 38fs) Less injection at higher power Relativistic effect, self-modulation 1 GeV beam: a0 ~ 1.46 (40 TW, 37 fs) Peak energy: 1000 MeV Divergence(rms): 2.0 mrad Energy spread (rms): 2.5% Resolution: 2.4% Charge: > 30.0 pC With 312 micron diameter, 33 mm length capillary, 1 GeV electron beam was generated. This is a spectrum. Input laser was 40 TW - 37 fs. Peak energy was 1 GeV, divergence was 2.0 mrad, energy spread was 2.5 %. Here, the resolution of the magnetic spectrometer was 2.4 %. This spread was clearly resolution limited, means might have been smaller. Charge was 30 pC. This results is published very recently from Nature Physics. Less stable operation Laser power fluctuation, discharge timing, pointing stability

Wake Evolution and Dephasing WAKE FORMING 200 Longitudinal Momentum Propagation Distance INJECTION 200 Longitudinal Momentum Propagation Distance 200 DEPHASING DEPHASING Longitudinal Momentum Propagation Distance Geddes et al., Nature (2004) & Phys. Plasmas (2005)

Bubble regime: Ultra-relativistic laser, I=1020 W/cm2 A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (2002) laser 12J, 33 fs trapped e- Z/ l -50 cavity 0 200 400 E, MeV t=350 t=450 t=550 t=650 t=750 t=850 5 10 8 1 10 9 N e / MeV Time evolution of electron spectrum