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Lecture 3: Laser Wake Field Acceleration (LWFA)

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. DLA. LWFA.

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Lecture 3: Laser Wake Field Acceleration (LWFA)

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  1. 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

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

  3. 0.2 0.2 eEz/wpmc wakefield breaks after few oscillations -0.2 eEz/wpmc -0.2 40 g 20 40 What drives electrons to g ~ 40 in zone behind wavebreaking? 2 g eEx/w0mc -2 20 20 px/mc laser pulse length -20 zoom 3 zoom Laser amplitude a0 = 3 a 3 -3 eEx/w0mc 20 -3 l 20 Transverse momentum p/mc >> 3 0 p/mc 0 px/mc -20 -20 270 280 Z / l 280 270 Z / l Laser pulse excites plasma wave of length lp= c/wp lp z

  4. dt p = e E + v  B dt p2/2 = e E  p = e E||p|| + e E p G 0 2x103 Gain due to transverse (laser) field: -2x103 0 103 G|| e   G = 2 e E pdt c   Gain due to longitudinal (plasma) field: G 0104 G|| = 2 e E|| p|| dt 0 104 G|| How do the electrons gain energy?

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

  6. cold plasma 1D Relativistic Plasma Equations (without laser) Consider an electron plasma with density N(x,t), velocity u(x,t), and electric fieldE(x,t), all depending on one spatial coordinate x and timet. Ions with densityN0 are modelled as a uniform, immobile, neutralizing background. This plasma is described by the 1D equations:

  7. 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

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

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

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

  11. 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 intwhen 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.

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

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

  14. 1D separatrix 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 cold fluid orbit (e- initially at rest)

  15. 0 acceleration range For maximum wave amplitude (in units,first obtained by Esarey, Piloff 1995) Maximum electron energy gain Wmaxin wakefield Electron acceleration (norm. quantities)

  16. single electron motion injected at phase velocity p/mc = bg Wave-Breaking at (bg)ph collective motion of plasma electrons 0 E/E0 Longitudinal E-field Wave Breaking p/mc = b

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

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

  19. 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 40-100 TW, 40 fs 10 Hz Laser: 3 cm

  20. 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) Peak energy: 490 MeV Divergence(rms): 1.6 mrad Energy spread (rms): 5.6% Resolution: 1.1% Charge: ~50 pC

  21. 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 Less stable operation Laser power fluctuation, discharge timing, pointing stability

  22. Wake Evolution and Dephasing 200 WAKE FORMING Longitudinal Momentum 0 Propagation Distance 200 INJECTION Longitudinal Momentum 0 Propagation Distance 200 DEPHASING DEPHASING Longitudinal Momentum 0 Propagation Distance Geddes et al., Nature (2004) & Phys. Plasmas (2005)

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

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