Post on 21-Aug-2020
Strong fields in laser plasmas- an introduction to high intensity laser plasma interaction -
Matthias Schn�rer: schnuerer@mbi-berlin.de
1 Laser acceleration experimentsExperimental techniques & measurement of strong fields
2 Electrons kinematicsand 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, MBIeuropean activities cf. e.g.: www.extreme-light-infrastructure.eu
referencesat the end
lecture is a compilationout of books, PhD-thesis,articles,(Don�t blame mefor footnotes )
1/15/2012 1M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption
Grading of ´´strong´´ , ´´high´´ ´´relativistic´´
particle gains energy in field => kinematics starts to become relativisticsay:T = (-1) m0 c2 = m0 c2 ( = 2)
= (1-2)-1/2 = v/c
electron in oscillatory E-field
me x = e E0 cos ( t)integrationv(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
0T (.) dt = e2 E20/(4 me e2 2) = Up
..
1/15/2012 2M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption
calculation concerning max. kinetic energy
E0 = 2 me c / e
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
a0 = 1 boundary a0 = vosc/c = 1
such high field strength can be produced with em-fields of lasersif 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
cf. atomic unitsfield strength~ 5 109 V/cm
cf. belowadditional associationwith a0
em-fieldwave equationsolution Poynting vector
1/15/2012 3M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption
Single electron interaction with an em – wave
Maxwell equationsrot E = - B/ t rot B = 00 j + E/ t div E = / 0 div B = 0
introduction of vector potential AE = - A/ t B = rot A
can derive wave equation, solution e.g.a propagating wave in x – direction expressed byA = ( 0, A0 cos (), (1 - 2)1/2 A0 sin ()) = t – k x = { +/- 1, 0} linear polarization = { +/- 1/Sqrt(2)} circular polarizationor e.g.Elin = E0 sin ( t – k x ) ey
one can derive an energy density of the em – field wemand can associate an energy current density wem c = S (Pointing vector)
which gives the relation to the intensity I (cycle averaged)
I = S = � 0 c E02
VECTOR
rot rot grad div - delta
simple analogyof wem in caseof electrostaticenergy densityof a capacitorW = � C U2
C = 0 A/dE = U/dFollowsWem = � 0 E2
1/15/2012 4M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption
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
plot example:
0 2 4 6 8 10 12 14 16 18 20 22
0.0
0.2
0.4
0.6
0.8
1.0
inte
nsity
cycle number
FWHM ~ 8 cycles ~ 22 fsin case of a 800 nm Ti:Sapph.laser
1/15/2012 5M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 6
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. previoussolution of relativistic case:trajectories:x= c a0
2/(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:
cf. references
P.Gibbon2005
PhD-thesisSokollik 2009Steinke 2010
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 7
PhD-thesisSokollik 2009Steinke 2010
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 8
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
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 9
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 treatmentand influence of B-field is omitted vy B -> 0, 2nd equation cancelled
-> integration first equation:
vy = e E0/ (me ω) sin (ωt – kx)
y = - vosc/ ω cos (ωt – kx)
vosc = e E0/ (me ω)
cf. caseinclusive Be.g. PhD-thesisNubbemeyer 2011
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 10
-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) with y = …
-> 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 dvy/dt dt !
0∫T cos(.) dt =0 0∫T cos2(.) dt = �
and vosc/ω e/me dE0/dy = e2/(me ω)2 E0 dE0/dy= � (e/(me ω))2 � d/dy (E0
2)
can associate
-> Fp = - � me (e/(me ω))2 d/dy (E02)
the ponderomotive force (act on a charge in an oscillating em-field)
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 11
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 = � 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 = - � (ωp/ ω) d(� ε0 E02)/dy = - ne/nc dUe/dy
orf = - � ne/nc Ue = - � ne me vosc
2 in comparisonto thermal pressure ptherm = (ne me ve
2)ponderomotive . exceeds thermal . for > 1015 W/cm2
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 12
Plasma – definition – examples
plasma term introduced by Langmuir in 1923, investigation of electrical dischargesmeans – something shaped – jelly like
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 13
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
thusδ/ δt (j) = e ne δ/ δt (ve) = e2 ne/me E
≡ e E/me
delivers (Ey only)
δ2/ δx2 (Ey) - ωp /c2 – � δ2/ δt2 (E) = 0
with ωp2 = ne e2 /(ε0 me) the plasma frequency
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 14
because wave equation has solution in form of
Ey = E0 exp( -i(kx – ωt))ifk2 = 1/c2 (ω2 – ωp
2) (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
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 15
Debye length
Plasma: quasi neutrality as a wholebut micro – fluctuation possible
Scale of possible charge fluctuations ?-> D ifelectrostatic 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 equationwhich gives for the electrostatic potential:
Uel ~ Z e/r exp(- r/ D )
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 16
Laser absorption in plasmas
collisional absorptiontreatment:ion – electron collision frequency eiintroduce in dispersion relation
ωL2 = ωp
2 ( 1 – i ei/ ωL ) k2 c2
gives 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)
use expression
ei = e4 Zav ne ln / (3 (2 )3/2 ε02 me
1/2 (kB Te)3/2
ei = 3 10-6 Zav ne[cm-3] ln / (Te [eV])3/2 s-1
Attwood
cf. relevantdefinitions
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 17
Coulomb logarithm ln ln = ln (bmin/bmax) is ratio of impact parameter
schematic picture
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] / (Zav
2 ne [cm-3]))
and for the local absorption kim(ei)
kim ~ ne2 Zav/(Te
3/2 (1 – ne/nc)1/2)
PhD-thesisSteinke 2010
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 18
Knowledge of ei gives also accessto dielectric constant(and npl – refraction coefficient of plasma)
= 1 – ne/nc 1 / (1 + i ei/ωL)
Have to determine the absorption coefficient globallyi.e. laser produced plasmas have density variations to consider
example: exponential density gradientne = nc exp (- x / L) , case L >> D
absorption s-polarized light
AL,s = 1 – exp ( - 8 ei(nc) L / (3 c) cos3 ())
is a slowly varying function with - incidence angle of laser
Kruer
p-polaricedcf. Gibbon
Homework:Why doesabsorption increase with higherlaserfrequency ?
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 19
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
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 veff
4)
veff = ( ve2 + vquiver
2)1/2
thus ei decreases and
keff(IL) = kim (1 + IL(3/(2 c nc kB Te)))-1
Elizier
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 20
When does collisionless absorption “overtake” collisional absorption ?
can estimate with “energy” comparison of the electron movementor “field” oscillation * coll.rate versus “thermal” movement * coll.rate
follows:
vquiver2 ei > Zav ei ve_therm
2
ifZav
-1 (vquiver2/ ve_therm
2 )= IL/ (Zav c nc kB Te) = 2.1 10-13 IL[W/cm-2] 2 [μm] / (Zav Te [eV])
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 21
Collisional absorption processes
Resonance absorption
Situation:P-polarized wave incident at angle target surface with plasma and density gradient L
ne = nc x/L
scheme
functional dependence:fa = Iabs/IL = 36 2 Ai()/(d Ai()/ d) = (k L)1/3 sin()
plot
principle:resonant excitationof plasma waveswith p = Lat ne = nc,“tunneled” fractionof laser lightto nc position
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 22
Brunel absorptioncf. PRL 1987
situation:electron excursion lengthxp ~ vosc/ωL> L (scale length of density gradient)resonance condition at nc breaks downbutelectron 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
simple Brunel model triggered a lot of refined studies
PhD-thesisSokollik
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 23
Relativistic j B heating ponderomotive acceleration
if a0 >1B – field contributes to ponderomotive force Fp (term for ponderomotive force is similar as deduced above)
2 effectsFp pushes electrons out of the focal regionelectrons 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
reason:E, B Interrelation
in principle:expression- heating -probably notexactbetter:change of electrondistributionfunction
1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 24
Anomalous skin effect
for ne > nc decaying E-fieldscale is skindepth
ls = c/ωp
(example: 800 nm laser ωL 2.35 1015 s-1, ne = 100 nc, thus ωp = 10 ωLand ls ~ 13 nm , or lower if layer is highly ionized)
anomalous means:mean free path between collisions > lsorvth / ei ~ lc > ls vth = ( kB Te/ me)1/2
description of extended layer lc (the anomalous skin layer) in which absorption takes placelc ~ (vth c2 /(ω ωp
2)1/3
and relative absorptionasa ~ (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
different opinionsconcerningsimilarityto Brunel
1/15/2012M. Schn�rer Strong fields in laser plasmas 2– Electron kinematics and laser absorption 25
ftp://ftp.mbi-berlin.de/pub/users/mschnue/ASP_Jena2012/
BooksShalom ElizierThe interaction of high-power lasers with plasmasIoP Publishing, 2002Paul GibbonShort pulse laser interaction with matterImperial College Press, 2005David AttwoodSoft X-rays and Extreme Ultaviolet RadiationCambridge University Press, 1999
ThesisThomas Sokollikhttp://opus.kobv.de/tuberlin/volltexte/2008/2028/pdf/sokollik_thomas.pdfSven Steinkehttp://opus.kobv.de/tuberlin/volltexte/2011/2885/pdf/steinke_sven.pdfAndreas Henighttp://edoc.ub.uni-muenchen.de/11483/1/Henig_Andreas.pdfRainer Hörleinhttp://edoc.ub.uni-muenchen.de/9615/1/Hoerlein_Rainer.pdfSebastian Pfotenhauerhttp://www.physik.uni-jena.de/qe/Papier/Dissertationen/Diss_Pfotenhauer.pdfOliver Jäckelwww.physik.uni-jena.de/inst/polaris/Publikation/Papier/Dissertationen/Diss_Jaeckel.pdfThomas Nubbemeyerhttp://opus.kobv.de/tuberlin/volltexte/2010/2723/pdf/nubbemeyer_thomas.pdf