Lecture 2 Basic plasma equations selffocusing direct laser

Lecture 2: Basic plasma equations, self-focusing, direct laser acceleration Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996) plasma box (ne/nc=0. 6) Lase r pul se 10 19 W/c 2 m B ~ mcwp/e ~ 108 Gauss Relat ivistic electr j ~ en on be 10 k. A c c ~ 1 12 0 A/ am of 1 -2 0 Me. V cm 2 electr ons 1

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) 2

Single electron plasma (ncrit = 1021 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. 3

The Virtual Laser Plasma Laboratory Three-dimensional electromagnetic fully-relativistic Particle-Cell-Code A. Pukhov, J. Plas. Phys. 61, 425 (1999) Fields Particles 109 particles in 108 grid cells are treated on 512 Processors of parallel computer 4

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

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

Problem: Derive cold plasma electron fluid equation 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 f 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 7

Basic solution of Solution for electron fluid initially at rest, before hit by laser pulse, implying balance between the electrostatic force ponderomotive force and the 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. 8

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

Relativistic Non-Linear Optics Induced transparency: w 2 = wp 2 + c 2 k 2 wp 2= 4 pe 2 ne /(m<g>) g =(1 - v 2/c 2)-1/2 n. R = (1 - wp 2/ w 2)1/2 Self-focussing: vph= c/n. R Profile steepening: vg = cn. R 10

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 n. R is the plasma index of refraction 11

3 D-PIC simulation of laser beam selffocussing in plasma Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996) plasma box (ne/nc=0. 6) Lase 10 19 r pul se W/c 2 m 12

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=z-ct, neglect second derivatives): 13

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. 14

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. 15

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

3 D-PIC simulation of laser beam selffocussing in plasma Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996) plasma box (ne/nc=0. 6) Lase r pul se 10 19 W/c 2 m B ~ mcwp/e ~ 108 Gauss Relat ivistic electr j ~ en on be 10 k. A c c ~ 1 12 0 A/ am of 1 -2 0 Me. V cm 2 electr ons 17

Relativistic self-focussing of laser channels wp 2= 4 p e 2 ne / mg eff w p 2 g ne radius relativistic electrons Relativistic mass increase (g ) and electron density depletion (ne ) increases index of refraction in the channel region, leading to selffocussing B-field laser 18

Relativistic Laser Plasma Channel Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996) B IL jx Intensity 80 fs ne ne/<g> B-field Intensity 330 fs Ion density 19

Plasma channels and electron beams observed C. Gahn et al. PRL 83, 4772 (1999) laser gas jet 6× 1019 W/cm 2 plasma 1 - 4 × 1020 cm-3 electron spectrum observed channel 20

Scaling of Electron Spectra Pukhov, Sheng, Mt. V, Phys. Plasm. 6, 2847 (1999) Teff =1. 8 (Il 2/13. 7 GW)1/2 electrons 21

Direct Laser Acceleration versus Wakefield Acceleration DLA electron B LWFA laser Non-linear plasma wave E plasma channel acceleration by transverse laser field Free Electron Laser (FEL) physics Pukhov, Mt. V, Sheng, Phys. Plas. 6, 2847 (1999) acceleration by longitudinal wakefield Tajima, Dawson, PRL 43, 267 (1979)22

Laser pulse excites plasma wave of length lp= c/wp 0. 2 lp 0. 2 e. Ez/wpmc -0. 2 40 -0. 2 g wakefield breaks after few oscillations laser pulse length 20 2 40 e. Ex/w mc g 0 What drives electrons to g ~ 40 in zone behind wavebreaking? -2 20 20 px/mc -20 a e. Ex/w 0 mc px/mc p /mc 3 3 -3 -3 2020 z zoom Laser amplitude a 0 = 3 l 00 -20 270 Transverse momentum p /mc >> 3 Z /l 280 23

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

How do the electrons gain energy? 2 x 103 Long pulses (> lp) G dt p = e E + e c v B Direct Laser Acceleration (long pulses) 0 dt p 2/2 = e E p = e E|| p|| + e E p -2 x 103 Gain due to transverse (laser) field: 0 G|| 103 G|| = 2 e E|| p|| dt 0 Gain due to longitudinal (plasma) field: Laser Wakefield Acceleration (short pulses) G G = 2 e E p dt 104 Short pulses (< lp) 0 G|| 4 1025

Selected papers: 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). C. Gahn, et al. Phys. Rev. Lett. 83, 4772 (1999). A. Pukhov, Z. M. Sheng, Meyer-ter-Vehn, Phys. Plasmas 6, 2847 (1999) 26

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=z-ct, neglect second derivatives): 27

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. 28

Problem: Derive channel fields B E space charge n = e(1 -f)n 0 j = efn 0 c 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 = ef. N 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, 29