strong fields in laser plasmas - an introduction to high ...2012_01_20_course2.pdf · photonic -...

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Strong fields in laser plasmas - an introduction to high intensity laser plasma interaction - Matthias Schnrer: [email protected] 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 references at the end lecture is a compilation out of books, PhD-thesis, articles, (Dont blame me for footnotes ) 1/15/2012 1 M. Schnrer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption

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Page 1: Strong fields in laser plasmas - an introduction to high ...2012_01_20_course2.pdf · Photonic - Laser books-> consequences of field gradients. M. Schn•rer Strong fields in laser

Strong fields in laser plasmas- an introduction to high intensity laser plasma interaction -

Matthias Schn�rer: [email protected]

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

Page 2: Strong fields in laser plasmas - an introduction to high ...2012_01_20_course2.pdf · Photonic - Laser books-> consequences of field gradients. M. Schn•rer Strong fields in laser

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

Page 3: Strong fields in laser plasmas - an introduction to high ...2012_01_20_course2.pdf · Photonic - Laser books-> consequences of field gradients. M. Schn•rer Strong fields in laser

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

Page 4: Strong fields in laser plasmas - an introduction to high ...2012_01_20_course2.pdf · Photonic - Laser books-> consequences of field gradients. M. Schn•rer Strong fields in laser

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

Page 5: Strong fields in laser plasmas - an introduction to high ...2012_01_20_course2.pdf · Photonic - Laser books-> consequences of field gradients. M. Schn•rer Strong fields in laser

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

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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

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1/15/2012M. Schn�rer Strong fields in laser plasmas 2 – Electron kinematics and laser absorption 7

PhD-thesisSokollik 2009Steinke 2010

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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

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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

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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)

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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

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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

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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

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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

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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 )

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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

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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

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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 ?

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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

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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])

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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

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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

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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

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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

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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