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Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)

plasma box ( n e /n c =0.6). Laser pulse 10 19 W/cm 2. B ~ mc w 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.

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Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)

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  1. plasma box (ne/nc=0.6) Laser pulse 1019 W/cm2 B ~ mcwp/e ~ 108 Gauss Relativistic electron beam j ~ encc ~ 1012 A/cm2 10 kA of 1-20 MeV electrons Lecture 2: Basic plasma equations, self-focusing, direct laser acceleration Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)

  2. Non-neutral plasma ( , usually fixed ion background) 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)

  3. They are quasi-stationary and of same order as laser fields: Single electron plasma (ncrit = 1021cm-3) • In plasma, laser interaction generates additional • E-fields (due to separation of electrons from ions) • B-fields (due to laser-driven electron currents) Plasma is governed by collective oscillatory electron motion.

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

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

  6. and assuming that only electrons with density Ne contribute to the plasma current while immobile ions with uniform density Ni=N0/Zform a neutralizing background. using normalized quantities and plasma frequency derive Problem: Light waves in plasma Starting from Maxwell equations

  7. 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

  8. 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 Basic solution of 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. Propagation of laser light in plasma 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 g – factor, giving rise to effects of relativistic non-linear optics.

  10. Induced transparency: Self-focussing: vph= c/nR Profile steepening: vg = cnR Relativistic Non-Linear Optics w2 = wp2 + c2k2 wp2= 4pe2 ne/(m<g>) g =(1- v2/c2)-1/2 nR = (1 - wp2/ w2)1/2

  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. plasma box (ne/nc=0.6) Laser pulse 1019 W/cm2 3D-PIC simulation of laser beam selffocussing in plasma Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)

  13. Problem: Derive envelope equation Consider circularly polarized light beam Confirm that the squared amplitude depends only on the slowly varying envelope function a0(r,z,t), but not on the rapidly oscillating phase function Derive under these conditions the envelope equation for propagation in vacuum (use comoving coordinatez=z-ct, neglect second derivatives):

  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. 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. Relativistic self-focusing

  16. Relativistic self-focusing develops differently in 2D and 3D geometry. Scaling with beam radius R : relativistic non-linearity diffraction 2D leads to a finite beam radius (R~1/P), while 3D leads to beam collapse (R->0). For a Gaussian beam with radius r0: power: beam radius evolution (Shvets, priv.comm.): critical power: 2D versus 3D relativistic self-focusing

  17. plasma box (ne/nc=0.6) Laser pulse 1019 W/cm2 B ~ mcwp/e ~ 108 Gauss Relativistic electron beam j ~ encc ~ 1012 A/cm2 10 kA of 1-20 MeV electrons 3D-PIC simulation of laser beam selffocussing in plasma Pukhov, Meyer-ter-Vehn, PRL 76, 3975 (1996)

  18. w p2 g wp2= 4p e2 ne/ mgeff ne radius relativistic electrons B-field laser Relativistic self-focussing of laser channels Relativistic mass increase (g ) and electron density depletion (ne ) increases index of refraction in the channel region, leading to selffocussing

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

  20. electronspectrum Plasma channels and electron beams observed C. Gahn et al. PRL 83, 4772 (1999) gas jet laser 6×1019 W/cm2 plasma 1- 4 × 1020 cm-3 observed channel

  21. Teff =1.8 (Il2/13.7GW)1/2 electrons Scaling of Electron Spectra Pukhov, Sheng, MtV, Phys. Plasm. 6, 2847 (1999)

  22. 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

  23. 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

  24. space charge n = e(1-f)n0 j = efn0c B Radial electron oscillations E electron momenta (wp/c) Channel fields and direct laser acceleration

  25. Long pulses (> lp) Direct Laser Acceleration (long pulses) dt p = e E + v  B dt p2/2 = e E  p = e E||p|| + e E p G 0 104 Gain due to transverse (laser) field: G 0 2x103 e   G = 2 e E pdt Short pulses (< lp) c   Laser Wakefield Acceleration (short pulses) 0 104 G|| -2x103 0 103 G|| Gain due to longitudinal (plasma) field: G|| = 2 e E|| p|| dt How do the electrons gain energy?

  26. 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)

  27. Problem: Derive envelope equation Consider circularly polarized light beam Confirm that the squared amplitude depends only on the slowly varying envelope function a0(r,z,t), but not on the rapidly oscillating phase function Derive under these conditions the envelope equation for propagation in vacuum (use comoving coordinatez=z-ct, neglect second derivatives):

  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. space charge n = e(1-f)n0 j = efn0c Consider an idealized laser plasma channel with uniform charge density N = e(1-f)N0c , 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 = efN0c. Show that the quasi-stationary channel fields are B E and that elctrons trapped in the channel l perform transverse oscillations at the betatron frequency, independent of f, Problem: Derive channel fields

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