1D Relativistic Plasma Equations (without laser) cold plasma 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 N 0 are modelled as a uniform, immobile, neutralizing background. This plasma is described by the 1D equations:

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

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  m. Then discuss the non-linear shapes qualitatively: Verify that the extrema of , n(  ), and the zeros of E(  ) do not shift in  when increasing  m, while the zeros of  (  ), n(  ), and the extrema of E(  ) are shifted such that velocity and density develop sharp crests, while the E-field acquires a sawtooth shape.

13. Problem: Maximum electron energy gain W max 0 acceleration range Verify energy gain for electron injected at phase Velocity according to drawing: Show that maximum energy for wavebreaking is

14. Problem: Verfy the Example E-field at wave-breaking: Plasma: Laser: Dephasing length: Required laser power:

Electron Trapping in the Broken-Wave Regime Plasma density: 3.5×10 19 /cm 3 Laser pulse : 6.6 fs 20 mJ 3 TW a 0 =1.7 Trapped electrons: colour ~ p z /mc Pukhov, MtV, Appl.Phys. B74, 355 ( 2002) Laser pulse 2000

Bubble regime: Ultra-relativistic laser, I=10 20 W/cm 2 : A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (2002) VLPL laser 12J, 33 fs trapped e - 0 Z/ -50 cavity E, MeV t=350 t=450 t=550 t=650 t=750 t= N e / MeV Time evolution of electron spectrum

The Bubble: Details A.Pukhov & J.Meyer-ter-Vehn, Appl. Phys. B, 74, p.355 (2002) 0 Z/ -50 T=500 T=700 laser 12J, 33 fs trapped e - cavity 700 Z / 500  (g)  -factor of electrons n e / n c a 2 eE z /mc  0 (f) (e) (d) Z/ Nonlinear laser compression down to a one-cycle pulse cavity wall e - beam accelerating field VLPL The trapping cross-section is  tr ~ 3  m 2

Mangles et al, Rutherford: 70 MeV beam Geddes et al, LBNL: 85 MeV beam Faure et al, LOA: 170 MeV beam Observation of monoenergetic electron beams Nature 431 (2004):

First observation of bubble acceleration J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J. Rousseau, F. Burgy, V. Malka, Nature 431, 541 (2004) Laser pulse: 1J, 30 fs He-gas n e = 6  cm -3 B=0 B>0 energy: 170 ± 20 MeV charge: 0.5 nC divergence: 10 mrad 10% laser->electron conv. n e = 2  cm -3

J. Faure, Y. Glinec, A. Pukhov, S. Kiselev, S. Gordienko, E. Lefebvre, J. Rousseau, F. Burgy, V. Malka, Nature 431, 541 (2004) Experiment Simulation (Pukhov) Electron spectrum peaked in energy The experimental spectrum is peaked in energy in good agreement with 3D-PIC simulation

04080 E (MeV) Electrons/MeV 2x10 9 electrons at 70 ± 10 MeV 15 % of laser energy 1 E z (TV/m) Bubble Acceleration (3D-PIC) M.Geissler, J. Schreiber, J. Meyer-ter-Vehn, New J. Phys. 8, 186 (2006).( n e (10 20 cm -3 ) Laser z (  m) Laser 5fs, 115 mJ plasma: 2 x10 19 cm -3 (a 0 =5) electron path

S. Gordienko, A. Pukhov, Phys. Plasmas 12, (2005)

3D-PIC movie (5 fs, 115 mJ,a 0 =5) M. Geissler (2005) M.Geissler, J. Schreiber, J. Meyer-ter-Vehn, New J. Phys. 8, 186 (2006).

a 0 = 3 a 0 =5 Intensity threshold for electron trapping in bubble M.Geissler, J. Schreiber, J. Meyer-ter-Vehn, NJP submitted (2006).

Number of electrons in peak > 60 MeV In bubble volume initially Energy in peak Laser energy Number and total energy of 70 MeV electrons extract beam

Particle trajectories: 5fs, a=5, 60-80MeV e -

“Monoenergetic” Energy Spectra from Regular Structures Regular accelerating structure  E =E(x 0 ) “Monoenergetic” spikes at stationary points Universal spectrum for continuous particle trapping For LWFA case see also T. Esirkepov et al., PRL, 2006 At a first-order maximum

“Monoenergetic” Energy Spectra from Stationary Points dN/dE EmEm E 1 st order maximum E EmEm x0x0 x 0m N E, MeV t=350 t=450 t=550 t=650 t=750 t= e

The bubble fields Kostyukov, Pukhov, Kiselev, Phys. Plasmas 11, 5256 (2004) Maxwellpotentialsspherical bubble: laboratory frame choose gauge solution: harmonic oscillator potential

The bubble fields E field: potentials: B field: Force on electron with velocity :

Consider a spherical bubble (radius R, n e = 0, n i =n 0 for r < R) moving in the lab frame in x-direction at velocity c. Show that the electric potential corresponds to a harmonic oscillator Start from Maxwell equations (no current inside bubble: ions static, no electrons!) with normalized quantities (confirm!) Use gauge for vector potential (why allowed ?). Derive the electric and the magnetic field inside the bubble. Show that an electron comoving with the bubble at velocity c experiences a force linear in bubble radius, while no transverse force acts on an electron moving in opposite direction (v=-ce x ). 12. Problem: Bubble fields (Kostyukov, Pukhov, Kiselev, Phys. Plasmas 11, 5256 (2004))

13. Problem: Ultra-relativistic laser plasma scaling S. Gordienko, A. Pukhov, Phys. Plasmas 12, (2005) rel. laser plasma equations Show that, in ultra-relativistic limit ( a 0 >>1 ), scaled. laser plasma equations depend only on three parameters:

Geometric Similarity, S=10 -3 =const (i) a2a ne/ncne/nc Y / a 0 =10 n e = 0.01n c (ii) a2a ne/ncne/nc a 0 =20 n e = 0.02n c (iii) a2a2 X / ne/ncne/nc a 0 =40 n e = 0.04n c (iv) a2a2 X / ne/ncne/nc a 0 =80 n e = 0.08n c Gordienko, Pukhov, Phys. Plasmas 12, (2005)

bubble regime scales with peak electron energy number of acc. electrons optimal focus laser-to-electron efficiency

Conventional accelerators 50 GeV LINAC at Stanford 3 km 30 km International Liner Collider: 500 GeV, 31 km, $6.7 bln The accelerating field is limited to a few 10 MeV/m

Present day accelerators are tens of kilometers long and reach their limits in size. The hope is that laser accelerators can be built with much smaller dimensions reaching higher energies

Accelerating fields Conventional RF accelerators E < 100 MV/m limit set by electrical breakdown Space charge fields, laser-generated E  mc  p /e < 100 TV/m for solid-density plasma E Laser W/cm 2

5 GeV protons at W/cm 2 1 kJ, 15 fs laser pulse focussed on 10  m spot of /cm 3 plasma protons GeV

Selected Publications: T. Tajima, J.M. Dawson, Phys.Rev.Lett. 43, 267 (1979) Pukhov & Meyer-ter-Vehn, Appl. Phys. B74, 355 (2002). S. Gordienko, A. Pukhov, Phys. Plasmas 12, (2005). Kostyukov, Pukhov, Kiselev, Phys. Plasmas 11, 5256 (2004). E.Easarey, P.Sprangle, J.Krall, A.Ting, IEEE Trans. Plas.Science 24, 252 (1966). Recent Survey and Collection of papers on: Laser-driven particle accelerators: new sources of energetic particles and radiation. Ed. K.Burnett, D. Jaroszynski, S. Hooker, Phil. Trans. Royal Soc. A 364, 551 – 778 (2006) Exp. Observation of laser-driven mono-energetic electron beams: Nature 431 (2004) Dream Beam Issue, 1.S.P.Mangles, C.D.Murphy, Z.Najmudin et al., 2.C.G. Geddes, E.Esarey, W.P.Leemans et al. 3. J. Faure, Y. Glinec, A. Pukhov, V. Malka, et al. F.S.Tsung, W. Lu, M.Tsoulas, W.B.Mori, C. Joshi, J.M. Vieira, L.O.Silva, R.A.Fonseca, Phys. Plasmas 13, (2006). M.Geissler, J. Schreiber, J. Meyer-ter-Vehn, New J. Phys. 8, 186 (2006).