1. 1 Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996) Laser pulse 10 19 W/cm 2 plasma box (n e /n c =0.6) B ~ mc  p /e ~ 10 8 Gauss Relativistic electron beam j ~ en c c ~ 10 12 A/cm 2 10 kA of 1-20 MeV electrons Lecture 2: Basic plasma equations, self-focusing, direct laser acceleration

2. 2 Laser Interaction with Dense Matter Plasma approximation: Laser field at a > 1 so large that atoms ionize within less than laser cycle Free classical electrons (no bound states, no Dirac equation) Non-neutral plasma (, usually fixed ion background)

3. 3 Single electron plasma (n crit = 10 21 cm -3 ) In plasma, laser interaction generates additional E-fields (due to separation of electrons from ions) B-fields (due to laser-driven electron currents) They are quasi-stationary and of same order as laser fields: Plasma is governed by collective oscillatory electron motion.

4. 4 The Virtual Laser Plasma Laboratory Fields Particles 10 9 particles in 10 8 grid cells are treated on 512 Processors of parallel computer A. Pukhov, J. Plas. Phys. 61, 425 (1999) Three-dimensional electromagnetic fully-relativistic Particle-Cell-Code

5. 5 Theoretical description of plasma dynamics Distribution function: Kinetic (Vlasov) equation ( ): Fluid description: Approximate equations for density, momentum, ect. functions:

6. 6 Starting from Maxwell equations Problem: Light waves in plasma and assuming that only electrons with density Ne contribute to the plasma current while immobile ions with uniform density N i =N 0 /Z form a neutralizing background. using normalized quantities and plasma frequency derive

7. 7 In this approximation, electrons are described as cold fluid elements which have relativistic momentum and satisfy the equation of motion where pressure terms proportional to plasma temperature have been neglected. Using again the potentials A and  and replacing the total time derivative by by partial derivatives, find and show that this leads to the equation of motion of a cold electron fluid written again in normalized quantities (see previous problem). Here, make use of Problem: Derive cold plasma electron fluid equation

8. 8 Basic solution of Solution for electron fluid initially at rest, before hit by laser pulse, implying balance between the electrostatic force and the ponderomotive force This force is equivalent to the dimensional force density It describes how plasma electrons are pushed in front of a laser pulse and the radial pressure equilibrium in laser plasma channels, in which light pressure expels electrons building up radial electric fields.

9. 9 For low laser intensities ( ), the solution implies and. The wave equation for laser propagation in plasma then leads to the plasma dispersion relation For increasing light intensity, the plasma frequency is modified by changes of electron density and relativistic  – factor, giving rise to effects of relativistic non-linear optics. Propagation of laser light in plasma

10. 10 Relativistic Non-Linear Optics Self-focussing: v ph = c/n R Profile steepening: v g = cn R  p 2 = 4  e 2 n e /(m<  n R = (1 -  p 2 /  2 ) 1/2  2 =  p 2 + c 2 k 2  =(1- v 2 /c 2 ) -1/2 Induced transparency:

11. 11 Problem: Derive phase and group velocity of laser wave in plasma Starting from the plasma dispersion relation show that the phase velocity of laser light in plasma is and the group velocity where nR is the plasma index of refraction

12. 12 3D-PIC simulation of laser beam selffocussing in plasma Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996) Laser pulse 10 19 W/cm 2 plasma box (n e /n c =0.6)

13. 13 Problem: Derive envelope equation Consider circularly polarized light beam Confirm that the squared amplitude depends only on the slowly varying envelope function a 0 (r,z,t), but not on the rapidly oscillating phase function Derive under these conditions the envelope equation for propagation in vacuum (use comoving coordinate   =z-ct, neglect second derivatives):

14. 14 Problem: Verify Gaussian focus solution Show that the Gaussian envelope ansatz inserted into the envelope equation leads to where is the Rayleigh length giving the length of the focal region.

15. 15 Relativistic self-focusing For increasing light intensity, non-linear effects in light propagation first show up In the relativistic factor giving and leads to the envelope equation (using !) While is defocusing the beam (diffraction), the term is focusing the beam. Beyond the threshold power the beam undergoes relativistic self-focusing.

16. 16 2D versus 3D relativistic self-focusing Relativistic self-focusing develops differently in 2D and 3D geometry. Scaling with beam radius R : diffraction relativistic non-linearity 2D leads to a finite beam radius ( R~1/P ), while 3D leads to beam collapse (R->0). For a Gaussian beam with radius r 0 : power: beam radius evolution (Shvets, priv.comm.): critical power:

17. 17 3D-PIC simulation of laser beam selffocussing in plasma Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996) Laser pulse 10 19 W/cm 2 plasma box (n e /n c =0.6) B ~ mc  p /e ~ 10 8 Gauss Relativistic electron beam j ~ en c c ~ 10 12 A/cm 2 10 kA of 1-20 MeV electrons

18. 18 Relativistic self-focussing of laser channels relativistic electrons laser B-field  p 2 radius  nene  p 2 = 4  e 2 n e / m   eff Relativistic mass increase (  ) and electron density depletion ( n e ) increases index of refraction in the channel region, leading to selffocussing

19. 19 Relativistic Laser Plasma Channel Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996) n e / nene Intensity B-field Intensity Ion density 80 fs 330 fs BB jxjx ILIL

20. 20 Plasma channels and electron beams observed C. Gahn et al. PRL 83, 4772 (1999) gas jet laser 6×10 19 W/cm 2 observed channel electron spectrum plasma 1- 4 × 10 20 cm -3

21. 21 Scaling of Electron Spectra Pukhov, Sheng, MtV, Phys. Plasm. 6, 2847 (1999) electrons T eff =1.8 ( I 2 /13.7GW) 1/2

22. 22 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

23. 23 Laser pulse excites plasma wave of length p = c/  p -0.2 0.2 eE z /  p mc 2 -2 eE x /  0 mc -20 20 p x /mc 40 20  eE x /  0 mc Z / 270 280 3 -3 -3 -3 -3 20 -20 0 p x /mc zoom -0.2 0.2 eE z /  p mc wakefield breaks after few oscillations 40 20  What drives electrons to  ~ 40 in zone behind wavebreaking? Laser amplitude a 0 = 3 Transverse momentum p  /mc >> 3 p  /mc zoom 3 -3 a 20 -20 0 Z / Z / 270 280 z laser pulse length p

24. 24 Channel fields and direct laser acceleration E B j = efn 0 c space charge n = e(1-f)n 0 Radial electron oscillations electron momenta  p /c)

25. 25 How do the electrons gain energy? d t p 2 /2 = e E  p = e E || p || + e E  p  d t p = e E + v  B e c  || = 2 e E || p || dt   Gain due to longitudinal (plasma) field:     = 2 e E  p  dt   Gain due to transverse (laser) field: -2x10 3 0 10 3  ||   0 2x10 3 Direct Laser Acceleration (long pulses) Long pulses (>  p ) 0 10 4  ||   0 10 4 Laser Wakefield Acceleration (short pulses) Short pulses (< p )

26. 26 Selected papers: C. Gahn, et al. Phys.Rev.Lett. 83, 4772 (1999). J. Meyer-ter-Vehn, A. Pukhov, Z.M. Sheng, in Atoms, Solids, and Plasmas In Super-Intense Laser Fields (eds. D.Batani, C.J.Joachain, S. Martelucci, A.N.Chester), Kluwer, Dordrecht, 2001. A. Pukhov, J. Meyer-ter-Vehn, Phys. Rev. Lett. 76, 3975 (1996). A. Pukhov, Z.M. Sheng, Meyer-ter-Vehn, Phys. Plasmas 6, 2847 (1999)

27. 27 Problem: Derive envelope equation Consider circularly polarized light beam Confirm that the squared amplitude depends only on the slowly varying envelope function a 0 (r,z,t), but not on the rapidly oscillating phase function Derive under these conditions the envelope equation for propagation in vacuum (use comoving coordinate   =z-ct, neglect second derivatives):

28. 28 Problem: Verify Gaussian focus solution Show that the Gaussian envelope ansatz inserted into the envelope equation leads to Where is the Rayleigh length giving the length of the focal region.

29. 29 Problem: Derive channel fields E B j = efn 0 c space charge n = e(1-f)n 0 Consider an idealized laser plasma channel with uniform charge density N = e(1-f)N 0 c, i.e. only a fraction f of electrons is left in the channel after Expulsion by the laser ponderomotive pressure, and this rest is moving With velocity c in laser direction forming the current j = efN 0 c. Show that the quasi-stationary channel fields are and that elctrons trapped in the channel l perform transverse oscillations at the betatron frequency, independent of f,