Lecture 2: Basic plasma equations, self-focusing, direct laser acceleration

Transcription

1 Lectue : Basic plasma equations, self-focusing, diect lase acceleation Pukhov, Meye-te-Vehn, PRL 76, 3975 (1996) plasma box (ne/nc=0.6) Lase pulse W/cm B ~ mcωp/e ~ 108 Gauss Relat ivistic elect j~e on be n cc ~ ka 0 A/ am of 1- cm 0 Me V ele cton s 1

2 Lase Inteaction with Dense Matte Plasma appoximation: Lase field at a > 1 so lage that atoms ionize within less than lase cycle Fee classical electons (no bound states, no Diac equation) Non-neutal plasma ( nelecton nion, usually fixed ion backgound)

3 Single electon plasma (ncit = 101cm-3) In plasma, lase inteaction geneates additional E-fields (due to sepaation of electons fom ions) B-fields (due to lase-diven electon cuents) They ae quasi-stationay and of same ode as lase fields: EL V/m a0 BL 108 Gauss a0 Plasma is govened by collective oscillatoy electon motion. 3

4 The Vitual Lase Plasma Laboatoy Thee-dimensional electomagnetic fully-elativistic Paticle-Cell-Code A. Pukhov, J. Plas. Phys. 61, 45 (1999) Fields Paticles ( 1 E 4π ot B = + j c t c dp q = qe + p B dt mγ 1 B ot E = c t p γ = 1+ mc div E = 4πρ div B = 0 ) 109 paticles in 108 gid cells ae teated on 51 Pocessos of paallel compute 4

5 Theoetical desciption of plasma dynamics Distibution function: Kinetic (Vlasov) equation ( t +v f (, p, t ) p = γ mv, γ = 1 + ( p / mc) e E + (v / c ) B ( ) ): f (, p, t ) 0 (collisions?) p Fluid desciption: Appoximate equations fo density, momentum, ect. functions: N (, t ) = f (, p, t ) d 3 p P (, t ) = p f (, p, t ) d 3 p 5

6 Poblem: Light waves in plasma Stating fom Maxwell equations E 4π =+ B == = J c t c B = E = = c t B 0 E A, B 4π e( N 0 Ne ) E A c t A 0 φ and assuming that only electons with density Ne contibute to the plasma cuent J = en e P / γ m with electon momentum P = γ mu and γ = 1 + ( P / mc), while immobile ions with unifom density Ni =N0/Z fom a neutalizing backgound. using nomalized quantities and plasma fequency ea eφ P Ne 4π e N 0 a (, t ) =, ϕ (, t ) =, p(, t ) =, n(, t ) =, ωp =, mc mc mc N0 m deive 1 =a c t ϕ + c t ω p np, c γ = ϕ (ω p / c )(n 1) 6

7 Poblem: Deive cold plasma electon fluid equation In this appoximation, electons ae descibed as cold fluid elements which have elativistic momentum P = γ mu and satisfy the equation of motion dp(, t ) / dt = e E + (u / c) B whee pessue tems popotional to plasma tempeatue have been neglected. Using again the potentials A and φ and eplacing the total time deivative by by patial deivatives, find A u + u= P+(, t ) e φ ( A) t c t c and show that this leads to the equation of motion of a cold electon fluid 1 ( p a ) =u c t ( p a) (ϕ γ ), witten again in nomalized quantities (see pevious poblem). Hee, make use of elations = γ+= = 1 p p / γ and u ( p) c γ (u )p. 7

8 Basic solution of 1 c t ( p a ) = u (p a) (ϕ γ) Solution fo electon fluid initially at est, befoe hit by lase pulse, p a and ϕ= γ implying balance between the electostatic foce pondeomotive foce =γ + 1 =p ϕ and the a / γ + 1 =a This foce is equivalent to the dimensional foce density ω p F = N 0 mc γ = γω E 8π It descibes how plasma electons ae pushed in font of a lase pulse and the adial pessue equilibium in lase plasma channels, in which light pessue expels electons building up adial electic fields. 8

9 Popagation of lase light in plasma ω p c na γ Fo low lase intensities ( a = 1 ), the p = a solution implies The wave equation fo lase popagation in plasma 1 =a c t γ 1 and n 1. ω p a, c then leads to the plasma dispesion elation ω = ω p + c k Fo inceasing light intensity, the plasma fequency is modified ω p,el 4π e N e (, t ) n = ωp = γ m γ (, t ) by changes of electon density and elativistic γ facto, giving ise to effects of elativistic non-linea optics. 9

10 Relativistic Non-Linea Optics Induced tanspaency: ω = ωp + ck ωp= 4πe ne /(m<γ>) γ =(1- v/c)-1/ nr = (1 - ωp/ ω)1/ Self-focussing: vph= c/nr Pofile steepening: vg = cnr 10

11 Poblem: Deive phase and goup velocity of lase wave in plasma Stating fom the plasma dispesion elation ω = ω p + c k, show that the phase velocity of lase light in plasma is v phase = ω / k = c / nr and the goup velocity vgoup = d ω / dk = cnr, whee nr is the plasma index of efaction nr = 1 ω p / ω. 11

12 3D-PIC simulation of lase beam selffocussing in plasma Pukhov, Meye-te-Vehn, PRL 76, 3975 (1996) plasma box (ne/nc=0.6) Lase pulse W/cm 1

13 Poblem: Deive envelope equation Conside ciculaly polaized light beam a = Re { (ey iez )a0 (, z, t ) exp(ikz iωt )} Confim that the squaed amplitude depends only on the slowly vaying envelope function a0(,z,t), but not on the apidly oscillating phase function a = a0 (, z, t ), a0 / t = ω a0 a0 / z = ka0 Deive unde these conditions the envelope equation fo popagation in vacuum (use comoving coodinate ζ=z-ct, neglect second deivatives): 1 =a c t 0 + ik ζ a0 (,=ζ ) 0 13

14 Poblem: Veify Gaussian focus solution Show that the Gaussian envelope ansatz a0 (, z ) = exp( P ( z ) + Q( z )( / 0 ) ) inseted into the envelope equation 1 + ik a0 (, z ) = 0 z leads to a0 (, z ) = whee e /[ 0 (1+ z / LR )] 1 + z / LR exp i actan z +i LR 0 z / LR 1 + z / LR LR = k0 / is the Rayleigh length giving the length of the focal egion. 14

15 Relativistic self-focusing Fo inceasing light intensity, non-linea effects in light popagation fist show up In the elativistic facto 1/ γ = 1/ 1 + a ; 1 a giving 1 = a c t ω p c na γ and leads to the envelope equation (using + ik z a0 ( =, z ) ω p 1 c a a, ω = ω p + c k!) ω p a0 a0 c While is defocusing the beam (diffaction), the tem (ω p / c )(a0 / )a0 a0 is focusing the beam. Beyond the theshold powe Pcit Po (ω / ω p ) = 17.4 GW (ncit / ne ) the beam undegoes elativistic self-focusing. 15

16 D vesus 3D elativistic self-focusing Relativistic self-focusing develops diffeently in D and 3D geomety. Scaling with beam adius R : diffaction elativistic non-lineaity : 1/ R (ω p / c )(a0 / ) : P / R (fo D: P : Ra0 ) (ω p / c )(a0 / ) : P / R (fo 3D: P : π R a0 ) D leads to a finite beam adius (R~1/P), while 3D leads to beam collapse (R->0). Fo a Gaussian beam with adius 0: powe: beam adius evolution (Shvets, piv.comm.): citical powe: P = π R I 0 / = P0 (ω /16c ) a0 R dr ( z ) 4 1 ωp = 3 1 a0 R dz k R 3 c Pcit Po (ω / ω p ) = 17.4 GW (ncit / ne ) 16

17 3D-PIC simulation of lase beam selffocussing in plasma Pukhov, Meye-te-Vehn, PRL 76, 3975 (1996) plasma box (ne/nc=0.6) Lase pulse W/cm B ~ mcωp/e ~ 108 Gauss Relat ivistic elect j~e on be n cc ~ ka 0 A/ am of 1- cm 0 Me V ele cton s 17

18 Relativistic self-focussing of lase channels ωp= 4π e ne / mγ eff ω p γ ne nr = 1 ω p ωl adius elativistic electons Relativistic mass incease (γ ) and electon density depletion (ne ) inceases index of efaction in the channel egion, leading to selffocussing B-field lase 18

19 Relativistic Lase Plasma Channel Pukhov, Meye-te-Vehn, PRL 76, 3975 (1996) B IL jx Intensity 80 fs ne ne/<γ> B-field Intensity 330 fs Ion density 19

20 Plasma channels and electon beams obseved C. Gahn et al. PRL 83, 477 (1999) lase gas jet W/cm plasma cm-3 electon spectum obseved channel 0

21 Scaling of Electon Specta Pukhov, Sheng, MtV, Phys. Plasm. 6, 847 (1999) Teff =1.8 (Iλ/13.7GW)1/ electons 1

22 Diect Lase Acceleation vesus Wakefield Acceleation DLA electon B LWFA lase Non-linea plasma wave E plasma channel acceleation by tansvese lase field Fee Electon Lase (FEL) physics Pukhov, MtV, Sheng, Phys. Plas. 6, 847 (1999) acceleation by longitudinal wakefield Tajima, Dawson, PRL43, 67 (1979)

23 Lase pulse excites plasma wave of length λp= c/ωp 0. λp eez/ωpmc 0. wakefield beaks afte few oscillations -0. eez/ωpmc γ lase pulse length 40 eex/ω mc 0 γ What dives electons to γ ~ 40 in zone behind wavebeaking? px/mc -0 a eex/ω0mc px/mc p /mc zoom zoom z Lase amplitude a0 = 3 λ Tansvese momentum p /mc >> Z / λ Z /80 λ 80 3

24 Channel fields and diect lase acceleation ee = (1 f )mω p R / ebϕ = f mω p R / d R mγ = ee ebϕ = mω p R / dt B E space chage n = e(1-f)n0 j = efn0c Radial electon oscillations Ω = ω p / γ ωl electon momenta ωl (ωp/c) 4

25 How do the electons gain enegy? x103 Long pulses (> λp) Γ dt p = e E + e c v B Diect Lase Acceleation (long pulses) 0 dt p/ = e E p = e E p + e E p -x103 Gain due to tansvese (lase) field: 0 Γ 103 Shot pulses (< λp) 0 Γ = e E p dt Lase Wakefield Acceleation (shot pulses) Γ Gain due to longitudinal (plasma) field: 104 Γ = e E p dt 0 Γ 4 105

26 Selected papes: J. Meye-te-Vehn, A. Pukhov, Z.M. Sheng, in Atoms, Solids, and Plasmas In Supe-Intense Lase Fields (eds. D.Batani, C.J.Joachain, S. Matelucci, A.N.Cheste), Kluwe, Dodecht, 001. A. Pukhov, J. Meye-te-Vehn, Phys. Rev. Lett. 76, 3975 (1996). C. Gahn, et al. Phys.Rev.Lett. 83, 477 (1999). A. Pukhov, Z.M. Sheng, Meye-te-Vehn, Phys. Plasmas 6, 847 (1999) 6

27 Poblem: Deive envelope equation Conside ciculaly polaized light beam a = Re { (ey iez )a0 (, z, t ) exp(ikz iωt )} Confim that the squaed amplitude depends only on the slowly vaying envelope function a0(,z,t), but not on the apidly oscillating phase function a = a0 (, z, t ), a0 / t = ω a0 a0 / z = ka0 Deive unde these conditions the envelope equation fo popagation in vacuum (use comoving coodinate ζ=z-ct, neglect second deivatives): 1 =a c t 0 + ik ζ a0 (,=ζ ) 0 7

28 Poblem: Veify Gaussian focus solution Show that the Gaussian envelope ansatz a0 (, z ) = exp( P ( z ) + Q( z )( / 0 ) ) inseted into the envelope equation 1 + ik a0 (, z ) = 0 z leads to a0 (, z ) = e /[ 0 (1+ z / LR )] 1 + z / LR exp i actan z +i LR 0 z / LR 1 + z / LR Whee LR = k0 / is the Rayleigh length giving the length of the focal egion. 8

29 Poblem: Deive channel fields B E space chage n = e(1-f)n0 j = efn0c Conside an idealized lase plasma channel with unifom chage density N = e(1-f)n0c, i.e. only a faction f of electons is left in the channel afte Expulsion by the lase pondeomotive pessue, and this est is moving With velocity c in lase diection foming the cuent j = efn0c. Show that the quasi-stationay channel fields ae ee = (1 f )mω p R /, ebϕ = f mω p R / and that elctons tapped in the channel l pefom tansvese oscillations at the betaton fequency, independent of f, Ω = ω p / γ 9