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 = 00 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