Relativistic laser beam propagation and critical density increase in a plasma Su-Ming Weng Theoretical Quantum Electronics (TQE), Technische Universität Darmstadt, Germany Joint work with Prof. Peter Mulser Moscow, 20 21 May 2010 3rd EMMI Workshop on Plasma Physics with Intense Laser and Heavy Ion Beams
Preface 1 Open Sesame, Ali baba and the forty thieves One Thousand and One Nights
Preface 2 Open Sesame, Ali baba and the forty thieves One Thousand and One Nights Who open the door for ultrahigh intense laser into an overdense plasma?
Outline 3 Theoretical background Classical eletromagnetic (EM) wave propagation Relativistic induced transparency Numerical simulations Relativistic critical density increase Relativistic laser beam propagation Potential applications Fast ignition Relativistic plasma shutter Shortening of laser pulses Conclusion
Classical EM wave propagation Maxwell s Equations 1 B E = c t 1 E 4π B= + J c t c For a high-frequency monochromatic laser beam in a plasma B t iω = iωb = E J = E t 4πω 2 E p 2 2, iω,, with ωp = 4 πe n / me, Maxwell s Equations iω E= B c iω B= ε E c 2 2 where ε=1- ωpe / ω is the dielectric function of the plasma. Eq. (1)-(2) 2 2 ω E E + 2 εe= (1) (2) current density ( ) 0 (wave equati on) c 4 plasma frequency
Classical EM wave propagation Dispersion relation E = 2 2 ω 0 holds for a uniform plasma in wave equation E ( E) + 0, 2 ε E = and assume that E exp( ik x iωt) is a plane wave ω 2 ε 2 2 2 2 2 k ω ω p c k, dispersion relation 2 = = + ( ) c plasma frequency ω is the minimum frequency for EM wave propagation in a plasma. p the electrons will shield out the EM field when ω < ω p c 5 Critical density the condition ω = ω defines the so-called critical density n, p n = mω /4πe = 1.1 10 / λ cm c e 2 2 21 2 3 Group velocity (propagation velocity) 1/2 1/2 2 vg 1 ω ω p n = ε = 1 1 2 = c c k ω nc c
Relativistic induced transparency 6 Dimensionless laser amplitude a: 2 π 2 18 W 2 Iλ = ca = 1.37 10 μm a2 2 2 cm Single particle s 8-like motion for a 1 y y x x-x d a<<1 a 1 T. C. Pesch and H. J. Kull, Phys. Plasmas 14, 083103 (2007).
Relativistic induced transparency 7 If v ~ c, m = (1 v / c ) m =γ m Relativistic critical density 2 2 1/2 e e0 e0 the Lorentz factor averaged from the single particle s 8-like motion 2 1/2 γ [1 + a / 2] ( Linear Polarization ), γ n = mω /4πe = γ n 2 2 cr e c [1 + a ] 2 1/2 Group velocity (relativistic) ( Circular Polarization ). 1/2 vg n n = 1 = 1 c n cr γ n c P. Mulser and D. Bauer, HighPowerLaser-Matter Interaction, (to be published by Springer). 1/2
Particle-in-Cell (PIC) simulation 8 Motion equations (particles) dr u 2 2 =, γ = 1 + u / c dt γ du u ms = qs( E+ B), s = e, i dt γ PIC simulation cycle + Maxwell s equations (fields) 1 E = tb, E = ρ ε 0 1 1 B =, 0 2 te + j B = 2 c ε 0c R. Lichters, et al., LPIC++, (1997).
Determination of critical density Laser and plasma parameters 9 0, x < 5 λl ne = nmax, x > 20λL nmax exp [( x 20) / L], otherwise
Determination of critical density Laser and plasma parameters Cycle-averaged propagation appears very regular, and laser is mainly reflected at the critical surface as that in nonrelativistic regime n n n 2 2 e = cos θ cr = γ e cos θ c, 10 for a = 10, incident angle θ =0 Incident wave energy density E = ( F ) + ( G ) in + 2 2 Reflected wave energy density E = ( F ) + ( G re with 2 + 2 ± F = ( E ± B )/ 2 y ± G = ( E m B ) / 2 z ) z y
Critical density VS laser intensity 11 A very smoothed critical density can be achieved after being averaged over 10 cycles. for a = 10, incident angle θ =0
Critical density VS laser intensity 12 A very smoothed critical density can be achieved after being averaged over 10 cycles. In a normally incident and linearly polarized laser pulse, relativistic critical density increase is in a good agreement with the averaged value from the single particle s 8-like motion n n R a 2 1/2 cr / c = γ = [1 + (1 + ) /2], ( R reflectivity).
Effect of plasma density profile 13 For normal incident, if density scale length L>λ L, n cr is almost independent of density profile For a very steep and highly overdense plasma. n cr is strongly suppressed n cr is only about 1.5 for a = 5, and n e 0, x < 5λL = 100 nc, x > 5λL electric field at the surface and skin depth 1/ n 1/2 e
Effect of laser polarization 14 For circular polarization, a density ridge prevents the laser from further propagation and restricts the critical density increase for a = 5, θ = 0 (a) Linear Theoretically polarization γ =[1 + ( + Ra 2 1/2 1 ) / 2] =4.98, And from PIC, γ is about 4.55. (b) Circular polarization Theoretically γ + + Ra 2 1/2 =[1 (1 ) ] =6.91, But from PIC γ is about 5.33.
Effect of laser polarization 15 For normal incident, the relativistic critical density increase can be well fitted by γ γ = [1 + 0.4a + 0.76 a ] = + + 0.62 2 1/ 2 (linear polarization) 1.2 2 1/ 2 [1 a 0.84 a ] (circular polarization)
Critical density VS incident angle The highest critical density coincides with the lowest reflectivity in the case of 30 o incidence. 2 1/2 conflicts with ncr / nc = γ = [1 + (1 + R) a / 2]. 16 for a = 2,
for a Critical density VS incident angle The highest critical density coincides with the lowest reflectivity in the case of 30 o incidence. 17 for a = 2, θ = 30 o This phenomenon is due to the formation of semiblack density cavity and resonance.
Relativistic laser beam propagation (LP) 18 linear polarization θ = 0, a = 10, at t=35 τ, and n e L 0, x < 5λL = 5 nc, x > 5λL Theoretically ( ) 1/2 v v = 1 n/ n c = 0.66c, prop g cr but from PIC v prop (15.5 5) λl = 0.35c, (35 5) τ L Sakagami and Mima attributed the inhibition of the propagation velocity to the oscillation of the ponderomotive force and hence the oscillation of electron density at the laser front. H. Sakagami, K. Mima, Phys. Rev. E 54, 1870 (1996).
Relativistic laser beam propagation (CP) Ponderomotive force for circular polarized laser m 2 f ( ) ˆ p = vos x x, vos ( x) = ee / mω, without oscillation 4 x 19 Theoretically ( ) 1/2 v v = 1 n/ n c = 0.7c, prop g cr but from PIC v prop (8.0 5) λl = 0.1c. (35 5) τ CP pulse propagates even more slowly than LP pulse. L Inhibition of propagation velocity is not attributed to the oscillation of ponderomotive force.
Relativistic laser beam propagation 20 Laser field penetrates into an overdense plasma for a = 5, and n e 0, x < 5λL = 10 0 nc, x > 5λL electric field at the surface and skin depth 1/ n 1/2 e Relativistic induced transparency is guided by the skin penetration. Propagation velocity depends on both the group velocity and the efficiency of skin penetration.
Relativistic propagation velocity 21 for a = 10, and θ = 0 (a) linear polarization vprop 2n 0 n 0 = exp 1 c γ n c γ n c 1/2 (b) circular polariztion a vprop 4n 0 n 0 = exp 1 c γ n c γ n c 1/2
Relativistic propagation velocity 22 for a = 10, and θ = 0 (a) linear polarization vprop 2n 0 n 0 = exp 1 c γ n c γ n c 1/2 (b) circular polariztion a vprop 4n 0 n 0 = exp 1 c γ n c γ n c different reduced factors ( n0 γ nc ) ( n0 γ nc ) exp 2 / (linear polarized) exp 4 / (circualr polarized) 1/2 the different heights of density ridge formed before the front of laser
Application (a): Fast ignition The configuration of fast ignition n e about 10 5 n c 23 Relativistic induced transparency, together with coneguiding, channeling and hole boring may make the deeper penetration into an overdense target possible, and make the fast ignition easier. J. J. Honrubia et al, Nucl. Fusion 46, L25 (2006)
Application (b): Relativistic plasma shutter 24 A relativistic plasma shutter can remove the pre-pulse and produce a clean ultrahigh intensity pulse S. A. Reed et al., Appl. Phys. Lett. 94, 201117 (2009). This shutter is overdense but relativistic underdense.
Application (c): Shortening of laser pulses 25 A quasi-single-cycle relativistic pulse can be produced by ultrahigh laser-foil interaction Initial thin overdense foil evolves into a thick but rare plasma, which is relativistically transparent. L. L. Ji et al., Phys. Rev. Lett. 103, 215005 (2009).
Conclusion 26 Relativistic induced transpancy make the penetration of a laser pulse into a overdense plasma possible. For normal incident linearly polarized pulse, the critical density increase approximately follows the theory from a single particle orbit. Due to the formation of a density ridge before the laser front, the critical density increase is much lower than predicted for a circular polarized pulse. Relativistic induced transparency is guided by the skin penetration; and the propagation velocity depends upon both the group velocity and the efficiency of skin penetration. Relativistic induced transparency could find wide applications in fast ignition scheme, relativistic plasma shutter, and shortening of laser pulses. Open Sesame! Open, skin penetration!
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