Strong fields in laser plasmas - an introduction to high intensity laser plasma interaction Matthias Schn€rer: schnuerer@mbi-berlin.de 1 Laser acceleration experiments Experimental techniques & measurement of strong fields 2 Electrons kinematics and laser absorption 3 Principles of laser – particle – acceleration - overview of some subjects concerning actual application of high intensity lasers - few basic considerations (not a consistent theoretical approach) - overview of experiments and techniques Centers in Germany: FSU, GSI, HHU, LMU-MPQ, MBI european activities cf. e.g.: www.extreme-light-infrastructure.eu M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption references at the end lecture is a compilation out of books, PhD-thesis, articles, (Don‚t blame me for footnotes ) 1/15/2012 1 Grading of ´´strong´´ , ´´high´´ ´´relativistic´´ particle gains energy in field => kinematics starts to become relativistic say: T = (-1) m0 c2 = m0 c2 ( = 2) = (1-2)-1/2 = v/c electron in oscillatory E-field .. me x = e E0 cos ( t) integration v(t) = e E0/(e ) sin ( t) max.kinetic energy (non.rel.) energy eqivalent of rest mass me /2 v2(t) = e2 E20/(2 me e2 2) sin2 ( t) = me c2 maximum energy: e2 E20/(2 me e2 2) = Umax cycle T ( = 2/ ) averaged energy T 0 (.) dt = e2 E20/(4 me e2 2) = Up M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 2 calculation concerning max. kinetic energy me c / e E0 = 2 relate = 2 f = 2 c / ( = 800 nm) E0 = 5.7 1010 V/cm call e E0 / (me c) = a0 dimensionless (normalized) field amplitude a0 > 1 region of relativistic kinematics cf. atomic units field strength ~ 5 109 V/cm cf. below additional association with a0 a0 = 1 boundary a0 = vosc/c = 1 such high field strength can be produced with em-fields of lasers if one associates E0 with a strong em-field of an intensity I one needs I 4.3 1018 W/cm2 or I 2.1 1018 W/cm2 (in case a0 = 1) E0 [ V/m] 27.4 I [W/cm2] => have to look to electron kinematics in ‚‚strong‚‚ em-fields M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption em-field wave equation solution Poynting vector 1/15/2012 3 Single electron interaction with an em – wave Maxwell equations rot E = - B/ t rot B = 00 j + E/ t div E = / 0 div B = 0 VECTOR introduction of vector potential A E = - A/ t B = rot A can derive wave equation, solution e.g. a propagating wave in x – direction expressed by A = ( 0, A0 cos (), (1 - 2)1/2 A0 sin ()) =t–kx = { +/- 1, 0} linear polarization = { +/- 1/Sqrt(2)} circular polarization or e.g. Elin = E0 sin ( t – k x ) ey one can derive an energy density of the em – field wem and can associate an energy current density wem c = S (Pointing vector) which gives the relation to the intensity I (cycle averaged) I = S = ƒ 0 c E02 M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption rot rot grad div - delta simple analogy of wem in case of electrostatic energy density of a capacitor W = ƒ C U2 C = 0 A/d E = U/d Follows W em = ƒ 0 E2 1/15/2012 4 useful relation a02 = I [W/cm2] 2 [m2] 0.73 10-18 plane wave -> pulse temporal function of field amplitude – envelope Eenv (t) e.g. field varies E(t) = Eenv (t) cos (t - t) Gauss: Eenv (t) = exp – (t/p)2 FWHM ~ 8 cycles ~ 22 fs in case of a 800 nm Ti:Sapph.laser 1.0 0.8 intensity plot example: 0.6 0.4 0.2 0.0 0 2 4 6 8 10 12 14 16 18 20 22 cycle number M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 5 Electron - field interaction - plane wave Lorentz equation: dp/ dt = d/dt (γme v) = -e (E + v B) non.-rel. case and v … B = 0 cf. previous solution of relativistic case: trajectories: x= c a02/(4ω) (Φ + (2δ2 – 1)/(2 sin(2 Φ))) y = δ c a0 /ω sin( Φ) z =- (1 - δ2)1/2/ ω cos(ω) example plots: M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption cf. references P.Gibbon 2005 PhD-thesis Sokollik 2009 Steinke 2010 1/15/2012 6 PhD-thesis Sokollik 2009 Steinke 2010 M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 7 need high intensity, high field strength plane wave -> focused wave -> spatial field distribution Example plot: focused gaussian beam with w0 = 5 beam waist parameter cf. references Photonic Laser books -> consequences of field gradients M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 8 Ponderomotive force - Electron in an em – wave (propagation along x, frequency ω, wavenumber k) non.rel. equations of motion: dvy/dt = e E0/ me cos (ωt – kx) dvx/dt = vy e B/ me cos (ωt – kx) - investigation: how does the spatial change of the field amplitude act on electron movement ( important situation -> field distribution in a focus) - look to a principle effect ( simplification – non-relativistic treatment and influence of B-field is omitted vy B -> 0, 2nd equation cancelled cf. case inclusive B e.g. PhD-thesis Nubbemeyer 2011 -> integration first equation: vy = e E0/ (me ω) sin (ωt – kx) y = - vosc/ ω cos (ωt – kx) vosc = e E0/ (me ω) M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 9 -introduce inhomogeneous field, description with a 2nd order term concerning E0 E0 -> E0 + y dE0/dy -> dvy/dt = e/me ( E0 + y dE0/dy) cos (ωt – kx) -> dvy/dt = e/me E0 cos (ωt – kx) - vosc/ω e/me dE0/dy cos2 (ωt – kx) ! look now to time - averaged effect 0∫ T cos(.) dt =0 with y = … T dvy/dt dt ! T cos2(.) dt = ƒ 0∫ 0∫ and vosc/ω e/me dE0/dy = e2/(me ω)2 E0 dE0/dy = ƒ (e/(me ω))2 ƒ d/dy (E02) can associate -> Fp = - ‹ me (e/(me ω))2 d/dy (E02) the ponderomotive force (act on a charge in an oscillating em-field) M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 10 Ponderomotive potential – useful relations - can associate Fp with the ponderomotive potential: Up ~ ∫ Fp dy ~ ‹ e2E02/(me ω2) ! ≡ ! cycle averaged oscillation energy - general relativistic treatment gives same relation ! (Up IL –relation) - with some plasma terms (cf. next considerations) Ue = ƒ ε0 E02 em-field energy density ωp = ne e2 / ε0 me nc = ωp2 ε0 me/ e2 one can write force per unit volume f = ne me dvy/dt = - ƒ (ωp/ ω) d(ƒ ε0 E02)/dy = - ne/nc dUe/dy or f = - ƒ ne/nc Ue = - ‹ ne me vosc2 in comparison to thermal pressure ptherm = (ne me ve2) ponderomotive . exceeds thermal . for > 1015 W/cm2 M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 11 Plasma – definition – examples plasma term introduced by Langmuir in 1923, investigation of electrical discharges means – something shaped – jelly like M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 12 Electromagnetic wave propagation in a plasma MWE in a medium -> wave equation ΔE – grad q/ε0 – μ0 δ/ δt (j) – 1/c2 δ2/ δt2 (E) = 0 c2 = 1/(ε0 μ0) ≡ 0 – plasma: quasi-neutrality of charge j = e ( ni vi – ne ve) consider only electrons because mi >> me and exclude ve gradients perpendicular to propagation thus δ/ δt (j) = e ne δ/ δt (ve) = e2 ne/me E ≡ e E/me delivers (Ey only) δ2/ δx2 (Ey) - ωp /c2 – ƒ δ2/ δt2 (E) = 0 with ωp2 = ne e2 /(ε0 me) the plasma frequency M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 13 because wave equation has solution in form of Ey = E0 exp( -i(kx – ωt)) if k2 = 1/c2 (ω2 – ωp2) (dispersion relation) important consequence: ω < ωp -> no wave propagation, only field penetration with exponential decay -> skin depth ( ~ c/ω) for ω = ωp -> ne = nc defines the critical density nc = ω2 ε0 me/e2 example: 800 nm laser , me = 512 keV , e = 1.6 10-19 As, ε0 = 8.85 10-12 As/Vm nc ~ 1.7 1021 cm-3 M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 14 Debye length Plasma: quasi neutrality as a whole but micro – fluctuation possible Scale of possible charge fluctuations ? -> D if electrostatic attraction force ( potential, pressure) ≡ thermodynamic force ( potential, pressure) intuitive estimation: Ne e2/(0 D ) = kBTe ne = Ne/ D3 D = (0 kBTe /(ne e2))1/2 example: Exact treatment requires solution of Poisson equation which gives for the electrostatic potential: Uel ~ Z e/r exp(- r/ D ) M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 15 Laser absorption in plasmas collisional absorption treatment: ion – electron collision frequency ei introduce in dispersion relation ωL2 = ωp2 ( 1 – i ei/ ωL ) k2 c2 gives Attwood k = kreal + i kim for a complex wave describes the term exp( - kim x ) -> absorption can write if ωL >> ei kim = ei ωp2 /(2 g ωL) = ei ne / ( 2 c nc (1 – ne/ nc)1/2) cf. relevant definitions use expression ei = e4 Zav ne ln / (3 (2 )3/2 ε02 me1/2 (kB Te)3/2 ei = 3 10-6 Zav ne[cm-3] ln / (Te [eV])3/2 s-1 M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 16 Coulomb logarithm ln ln = ln (bmin/bmax) is ratio of impact parameter schematic picture PhD-thesis Steinke 2010 can associate bmax -> D bmin -> coulomb energy Z e2/ (4 ε0 bmin) = thermal energy 3/2 kBTe gives approximation ln = 22.8 + 0.5 ln ( Te3 [eV] / (Zav2 ne [cm-3])) and for the local absorption kim(ei) kim ~ ne2 Zav/(Te3/2 (1 – ne/nc)1/2) M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 17 Knowledge of ei gives also access to dielectric constant (and npl – refraction coefficient of plasma) = 1 – ne/nc 1 / (1 + i ei/ωL) Have to determine the absorption coefficient globally i.e. laser produced plasmas have density variations to consider example: exponential density gradient ne = nc exp (- x / L) , case L >> D absorption s-polarized light Kruer p-polariced cf. Gibbon AL,s = 1 – exp ( - 8 ei(nc) L / (3 c) cos3 ()) is a slowly varying function with - incidence angle of laser M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption Homework: Why does absorption increase with higher laser frequency ? 1/15/2012 18 equipartition times timescale -> particle in equilibrium having Maxwell distribution tee = 0.29 (mec2)1/2 Te3/2 / (nec e4 ln) = 0.33 (Te[eV]/100)3/2 1021 / (ln () ne [cm-3]) ps relation tee tei tii Elizier e.g. tei ~ 1000 A/Zav2 tee Why do we have decreasing collisional absorption with increasing laser intensity IL? cross section e – i collisions: ei = 4 Zav e4 / (me2 veff4) veff = ( ve2 + vquiver2)1/2 thus ei decreases and keff(IL) = kim (1 + IL(3/(2 c nc kB Te)))-1 M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 19 When does collisionless absorption “overtake” collisional absorption ? can estimate with “energy” comparison of the electron movement or “field” oscillation * coll.rate versus “thermal” movement * coll.rate follows: vquiver2 ei > Zav ei ve_therm2 if Zav-1 (vquiver2/ ve_therm2 ) = IL/ (Zav c nc kB Te) = 2.1 10-13 IL[W/cm-2] 2 [μm] / (Zav Te [eV]) M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 20 Collisional absorption processes Resonance absorption Situation: P-polarized wave incident at angle target surface with plasma and density gradient L ne = nc x/L principle: resonant excitation of plasma waves with p = L at ne = nc, “tunneled” fraction of laser light to nc position scheme functional dependence: fa = Iabs/IL = 36 2 Ai()/(d Ai()/ d) = (k L)1/3 sin() plot M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 21 Brunel absorption cf. PRL 1987 situation: electron excursion length xp ~ vosc/ωL> L (scale length of density gradient) resonance condition at nc breaks down but electron acceleration into vacuum (half cycle) and acceleration back into plasma electrons reach region where em – wave is already strongly damped (ne > nc) electrons “leave” em – field “acquire” kinetic energy scheme for two cases a0 << 1 / 1 << a0 PhD-thesis Sokollik simple Brunel model triggered a lot of refined studies M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 22 Relativistic j B heating ponderomotive acceleration if a0 >1 B – field contributes to ponderomotive force Fp (term for ponderomotive force is similar as deduced above) 2 effects Fp pushes electrons out of the focal region electrons gain energy Ekin = Up = me c2 ( - 1) = me c2 ((1 + a02)1/2 – 1) force in x – direction (em – wave propagation) Fx = - me / 4 δ2/ δx2(vosc) ( 1 – cos2(ωLt)) pond. term oscillation in x due to j … B (electron trajectory) leads to “heating” effects are very close to action of light pressure plasma hole boring cf. lecture 2 : particle acceleration M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption reason: E, B Interrelation in principle: expression - heating probably not exact better: change of electron distribution function 1/15/2012 23 Anomalous skin effect for ne > nc decaying E-field scale is skindepth different opinions concerning similarity to Brunel ls = c/ωp (example: 800 nm laser ωL 2.35 1015 s-1, ne = 100 nc, thus ωp = 10 ωL and ls ~ 13 nm , or lower if layer is highly ionized) anomalous means: mean free path between collisions > ls or vth / ei ~ lc > ls vth = ( kB Te/ me)1/2 description of extended layer lc (the anomalous skin layer) in which absorption takes place lc ~ (vth c2 /(ω ωp2)1/3 and relative absorption asa ~ (kB Te/ (511 keV)1/6 (ne/nc)1/3 gives e.g. for ne = 100 nc and IL ~ 1019 W/cm-2 about 5% absorption M. Schn€rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 1/15/2012 24 ftp://ftp.mbi-berlin.de/pub/users/mschnue/ASP_Jena2012/ Books Shalom Elizier The interaction of high-power lasers with plasmas IoP Publishing, 2002 Paul Gibbon Short pulse laser interaction with matter Imperial College Press, 2005 David Attwood Soft X-rays and Extreme Ultaviolet Radiation Cambridge University Press, 1999 Thesis Thomas Sokollik http://opus.kobv.de/tuberlin/volltexte/2008/2028/pdf/sokollik_thomas.pdf Sven Steinke http://opus.kobv.de/tuberlin/volltexte/2011/2885/pdf/steinke_sven.pdf Andreas Henig http://edoc.ub.uni-muenchen.de/11483/1/Henig_Andreas.pdf Rainer Hörlein http://edoc.ub.uni-muenchen.de/9615/1/Hoerlein_Rainer.pdf Sebastian Pfotenhauer http://www.physik.uni-jena.de/qe/Papier/Dissertationen/Diss_Pfotenhauer.pdf Oliver Jäckel www.physik.uni-jena.de/inst/polaris/Publikation/Papier/Dissertationen/Diss_Jaeckel.pdf Thomas Nubbemeyer http://opus.kobv.de/tuberlin/volltexte/2010/2723/pdf/nubbemeyer_thomas.pdf M. Schn€rer Strong fields in laser plasmas 2– Electron kinematics and laser absorption 1/15/2012 25