laser plasma interactions physics of high...
TRANSCRIPT
Mitg
lied
derH
elm
holtz
-Gem
eins
chaf
t
Physics of High IntensityLaser Plasma InteractionsVarenna Summer School onLaser-Plasma Acceleration
20–25 June 2011 Paul Gibbon
Course outline
Lecture 1: Introduction – Definitions and Thresholds
Lecture 2: Interaction with Underdense Plasmas
Lecture 3: Interaction with Solids
Lecture 4: Numerical Simulation of Laser-Plasma Interactions
Lectures 5 & 6: Tutorial on Particle-in-Cell Simulation
2 133
Mitg
lied
derH
elm
holtz
-Gem
eins
chaf
t
Physics of High IntensityLaser Plasma InteractionsPart I: Definitions and thresholds
20–25 June 2011 Paul Gibbon
Lecture 1: Definitions and thresholds
Introduction
Field ionization
Relativistic threshold
Plasma Debye length
Plasma frequency
Further reading
Definitions and thresholds 4 133
Laser technology progress: chirped pulseamplification
Electronenergy
2
1970 1980 1990 2000 2010
1010
1510
10
2010
23
Year
1 keV
1 MeV
1 GeV
1 meV
CPA
1 eVMultiphoton physicsLaser medicine
Particle accelerationFusion schemes
ray sourcesγ−
Hot dense matter
Hard X−ray flash lamps
Field ionisation of hydrogen
Intensity (W/cm )
1960
Relativistic ions
Laser−nuclear physics
Relativistic optics: v ~ cosc
Figure 1: Progress in peak intensity since the invention of the laser.Definitions and thresholds Introduction 5 133
Extreme conditions: nonlinear, strong-field science
Ordinary matter — solid, liquid or gas — rapidly ionized whensubjected to high intensity irradiation.
Electrons released are then immediately caught in the laserfield
Oscillate with a characteristic energy which dictates thesubsequent interaction physics.
Basis for laser-based particle accelerator schemes andshort-wavelength radiation sources.
Definitions and thresholds Introduction 6 133
Electrons in intense electromagnetic fieldsprehistory
Volkov (1935): electron ‘dressed’ by field – relativistic massincrease
Schwinger (1949): cyclotron radiation
Invention of laser (1960): theoretical works on electrondynamics
Figure of merit q:
q =eEL
mωc, (1)
e = electron charge, m = electron mass, c = speed of light;EL = laser electric field strength; ω = light frequency.
Ostriker & Gunn (1969) – electron dynamics in vicinity ofpulsars: q ∼ 100
Definitions and thresholds Introduction 7 133
Field ionizationAt the Bohr radius
aB =~2
me2 = 5.3× 10−9 cm,
the electric field strength is:
Ea =e
4πε0a2B
' 5.1× 109 Vm−1. (2)
This leads to the atomic intensity :
Ia =ε0cE2
a
2' 3.51× 1016 Wcm−2. (3)
A laser intensity of IL > Ia will guarantee ionization for any targetmaterial, though in fact this can occur well below this thresholdvalue via multiphoton effects.
Definitions and thresholds Field ionization 8 133
Tunneling ionization
Keldysh (1965) and Perelomov (1966): introduced aparameter γ separating the multiphoton and tunnelingregimes, given by:
γ = ωL
√2Eion
IL∼
√Eion
Φpond
. (4)
where
Φpond =e2E2
L
4mω2L
(5)
is the ponderomotive potential of the laser field.
Regimes
γ < 1⇒ tunneling – strong fields, long wavelengthsγ > 1⇒ MPI – short wavelengths
Definitions and thresholds Field ionization 9 133
Tunnelling: barrier suppression model
��������������������
��������������������
���������
���������
V(x)
0
−E
x
xmax
ε x
ion
−e
Figure 2: a) Schematic picture of tunneling or barrier-suppressionionization by a strong external electric field.
Definitions and thresholds Field ionization 10 133
Barrier suppression model II
Coulomb potential modified by a stationary, homogeneouselectric field, see Fig. 13:
V (x) = −Ze2
x− eεx .
⇒ suppressed on RHS of the atom, and for x � xmax is lowerthan the binding energy of the electron.
If the barrier falls below Eion, the electron will escapespontaneously⇒ barrier suppression (BS) ionization.
Set dV (x)/dx = 0 to determine the position of the barrier:
xmax = (Ze/ε),
Definitions and thresholds Field ionization 11 133
Barrier suppression model III
Set V (xmax) = Eion to get the threshold field strength for BS:
εc =E2ion
4Ze3 . (6)
Equate critical field to the peak electric field of the laser –appearance intensity for ions created with charge Z :
Iapp =c
8πε2
c =cE4
ion
128πZ 2e6 , (7)
Appearance intensity
Iapp ' 4× 109(
Eion
eV
)4
Z−2 Wcm−2 (8)
Eion is the ionization potential of the ion or atom with charge(Z − 1).
Definitions and thresholds Field ionization 12 133
Appearance intensities of selected ions
Ion Eion Iapp
(eV) ( Wcm−2)
H+ 13.61 1.4× 1014
He+ 24.59 1.4× 1015
He2+ 54.42 8.8× 1015
C+ 11.2 6.4× 1013
C4+ 64.5 4.3× 1015
Ne+ 21.6 8.6× 1014
Ne7+ 207.3 1.5× 1017
Ar8+ 143.5 2.6× 1016
Xe+ 12.13 8.6× 1013
Xe8+ 105.9 7.8× 1015
Table 1: BS ionization model –Eq. (8).
Figure 3: Auguste et al., J. Phys. B(1992)
Definitions and thresholds Field ionization 13 133
Relativistic field strengths
Classical equation of motion for an electron exposed to a linearlypolarized laser field E = yE0 sinωt :
dvdt
' −eE0
mesinωt
→ v =eE0
meωcosωt = vos cosωt (9)
Dimensionless oscillation amplitude, or ’quiver’ velocity:
a0 ≡vosc≡ pos
mec≡ eE0
meωc(10)
Definitions and thresholds Relativistic threshold 14 133
Relativistic intensity
The laser intensity IL and wavelength λL are related to E0 and ωby:
IL =12ε0cE2
0 ; λL =2πcω
Substituting these into (10) we find (Exercise):
a0 ' 0.85(I18λ2µ)1/2, (11)
where
I18 =IL
1018 Wcm−2 ; λµ =λL
µm.
Implies that for IL ≥ 1018 Wcm−2, λL ' 1 µm, we will haverelativistic electron velocities, or a0 ∼ 1.
Definitions and thresholds Relativistic threshold 15 133
Plasma definitions: the Debye lengthA plasma created by field-ionization of a gas or solid will initiallybe quasi-neutral. This means that the number densities ofelectrons and ions with charge state Z are locally balanced:
ne ' Zni .
Any local disturbance in the charge distribution will be rapidlyneutralised by the lighter, faster electrons – Debye shielding:shields potential around exposed charge: φD = e−r/λD
r .
Debye length
λD =
(ε0kBTe
e2ne
)1/2
= 743(
Te
eV
)1/2( ne
cm−3
)−1/2
cm (12)
Definitions and thresholds Plasma Debye length 16 133
Plasma classificationAn ideal plasma has many particles per Debye sphere:
ND ≡ ne4π3λ3
D � 1. (13)
10 104 107 1010
Temperature (K)
102
106
1010
1014
1018
1024
1028
1032E
lect
ron
dens
ity(c
m-3
)
gaseous nebulaionosphere(F layer)
magneticfusion
electron gasin a metal
laser-plasmaslaser-plasmasJovian interior
stellar interiorstellar interior
solaratmosphere
solar corona
white dwarf
non-ideal p
lasmas
Figure 4: Ideal and non-ideal plasmasDefinitions and thresholds Plasma Debye length 17 133
Plasma frequency
For a thermal plasma with temperature Te one can also define acharacteristic reponse time (eg: to disturbances from externallaser fields or particle beams):
tD 'λD
vt=
(ε0kBTe
e2ne· m
kBTe
)1/2
=
(e2ne
ε0me
)−1/2
.
Electron plasma frequency
ωp ≡(
e2ne
ε0me
)1/2
' 5.6× 104(
ne
cm−3
)1/2
s−1. (14)
Definitions and thresholds Plasma frequency 18 133
Underdense and overdense plasmas
If the plasma response time is shorter than the period of aexternal electromagnetic field (such as a laser), then thisradiation will be shielded out.
Figure 5: Underdense, ω > ωp:plasma acts as nonlinearrefractive medium
Figure 6: Overdense, ω < ωp:plasma acts like mirror
Definitions and thresholds Plasma frequency 19 133
The critical density
To make this more quantitative, consider ratio:
ω2p
ω2 =e2ne
ε0me· λ2
4π2c2 .
Setting this to unity defines the wavelength for which ne = nc , orthe
Critical density
nc ' 1021λ−2µ cm−3 (15)
above which radiation with wavelengths λ > λµ will be reflected.cf: radio waves in ionosphere.
Definitions and thresholds Plasma frequency 20 133
Laser-generated plasmas
Target material Electron density ne ne/nc ( 800nm)( cm−3)
Capilliary discharge 1016 10−5
Gas jet 1018 10−3
Foam/aerogel 1021 0.1− 5Frozen H 1022 36CH foil 5× 1023 600
Table 2: Electron densities in typical laser-produced plasmas
Definitions and thresholds Plasma frequency 21 133
Further reading
IC Press, London (2005)
Definitions and thresholds Further reading 22 133
Summary of Lecture 1
Introduction
Field ionization
Relativistic threshold
Plasma Debye length
Plasma frequency
Further reading
Definitions and thresholds Further reading 23 133
Mitg
lied
derH
elm
holtz
-Gem
eins
chaf
t
Physics of High IntensityLaser Plasma InteractionsPart II: Interaction with UnderdensePlasmas
20–25 June 2011 Paul Gibbon
Lecture 2: Interaction with Underdense PlasmasPlasma responseCold fluid equations
EM wavesPlasma waves
EM wave propagationNonlinear refractionSelf focussingSF power thresholdPonderomotive channel formation
Plasma wave propagationDispersionNumerical solutionsWave breaking
Wakefield excitationElectron accelerationBench-Top Particle Accelerators
Interaction with Underdense Plasmas 25 133
Ionized gases: when is plasma response important?
Simultaneous field ionization of many atoms produces a plasmawith electron density ne, temperature Te ∼ 1− 10 eV. Collectiveeffects important if
ωpτL > 1
Example (Gas jet)
τL = 100 fs, ne = 1017 cm−3 → ωpτL = 1.8Typical gas jets: P ∼ 1bar; ne = 1018 − 1019 cm−3
Recall that from Eq.15, critical density for glass lasernc(1µ) = 1021 cm−3. Gas-jet plasmas are thereforeunderdense, since ω2/ω2
p = ne/nc � 1.
Exploit plasma effects for: short-wavelength radiation; nonlinearrefractive properties; high electric/magnetic fields.
Interaction with Underdense Plasmas Plasma response 26 133
Wave propagation
The starting point for most analyses of nonlinear wavepropagation phenomena is the Lorentz equation of motion forthe electrons in a cold (Te = 0), unmagnetized plasma, togetherwith Maxwell’s equations.
We also make two assumptions:
1 The ions are initially assumed to be singly charged (Z = 1)and are treated as a immobile (vi = 0), homogeneousbackground with n0 = Zni .
2 Thermal motion is neglected – justified for underdenseplasmas because the temperature remains small comparedto the typical oscillation energy in the laser field.
Interaction with Underdense Plasmas Cold fluid equations 27 133
Lorentz-Maxwell equations
Starting equations (SI units) are as follows
∂p∂t
+ (v · ∇)p = −e(E + v × B), (16)
∇·E =eε0
(n0 − ne), (17)
∇×E = −∂B∂t, (18)
c2∇×B = − eε0
nev +∂E∂t, (19)
∇·B = 0, (20)
where p = γmev and γ = (1 + p2/m2ec2)1/2.
Interaction with Underdense Plasmas Cold fluid equations 28 133
Electromagnetic waves
To simplify matters we first assume a plane-wave geometry likethat above. A laser pulse can thus be described by theelectromagnetic fields EL = (0,Ey , 0); BL = (0, 0,Bz).Exercise: From Eq. (16) one can show that the transverseelectron momentum is then simply given by:
py = eAy , (21)
where Ey = ∂Ay/∂t . This relation expresses conservation ofcanonical momentum.
Interaction with Underdense Plasmas Cold fluid equations EM waves 29 133
The EM wave equation I
Substitute E = −∇φ− ∂A/∂t ; B = ∇× A into Ampère Eq.(19):
c2∇× (∇× A) +∂2A∂t2 =
Jε0−∇∂φ
∂t,
where the current J = −enev .Now we use a bit of vectorial magic, splitting the current intorotational (solenoidal) and irrotational (longitudinal) parts:
J = J⊥ + J || = ∇×Π +∇Ψ
from which we can deduce (see Jackson!):
J || −1c2∇
∂φ
∂t= 0.
Interaction with Underdense Plasmas Cold fluid equations EM waves 30 133
The EM wave equation IINow apply Coulomb gauge ∇ · A = 0 and vy = eAy/γ from (21),to finally get:
EM wave
∂2Ay
∂t2 − c2∇2Ay = µ0Jy = − e2ne
ε0meγAy . (22)
The nonlinear source term on the RHS contains two importantbits of physics:
ne = n0 + δn → Coupling to plasma waves
γ =√
1 + p2/m2ec2 → Relativistic effects
Interaction with Underdense Plasmas Cold fluid equations EM waves 31 133
Electrostatic (Langmuir) waves I
Taking the longitudinal (x)-component of the momentumequation (16) gives:
ddt
(γmevx ) = −eEx −e2
2meγ
∂A2y
∂x
We can eliminate vx using Ampère’s law (19)x :
0 = − eε0
nevx +∂Ex
∂t,
while the electron density can be determined via Poisson’sequation (17):
ne = n0 −ε0
e∂Ex
∂x.
Interaction with Underdense Plasmas Cold fluid equations Plasma waves 32 133
Electrostatic (Langmuir) waves IIThe above (closed) set of equations can in principle be solvednumerically. For the moment, we simplify things by linearizingthe plasma quantities:
ne ' n0 + n1 + ...
vx ' v1 + v2 + ...
and neglect products like n1v1 etc. This finally leads to:
Driven plasma wave
(∂2
∂t2 +ω2
p
γ0
)Ex = −
ω2pe
2meγ20
∂
∂xA2
y (23)
The driving term on the RHS is the relativistic ponderomotiveforce, with γ0 = (1 + a2
0/2)1/2.
Interaction with Underdense Plasmas Cold fluid equations Plasma waves 33 133
Cold plasma wave equations: recap
Electromagnetic wave
∂2Ay
∂t2 − c2∇2Ay = µ0Jy = − e2ne
ε0meγAy
Electrostatic (Langmuir) wave
(∂2
∂t2 +ω2
p
γ0
)Ex = −
ω2pe
2meγ20
∂
∂xA2
y
These coupled fluid equations describe a vast range of nonlinearlaser-plasma interaction phenomena: parametric instabilities,self-focussing, channelling, wakefield excitation, harmonicgeneration, ...
Interaction with Underdense Plasmas Cold fluid equations Plasma waves 34 133
Dispersion properties: EM waves
This time we switch the plasma oscillations off (ne = n0) inEq.(22) and look for solutions:
Ay = A0 sin(ωt − kx),
to obtain
ω2 =ω2
p
γ0+ c2k2. (24)
From this relation we can derive a
Nonlinear refractive index
η =
√c2k2
ω2 =
(1−
ω2p
γ0ω2
)1/2
(25)
Interaction with Underdense Plasmas EM wave propagation 35 133
Nonlinear refraction effectsHave so far assumed plane wave approximation for laser pulse –’photon bullet’Real laser pulses are created with focusing optics & are subjectto:
1 diffraction due to finite focal spot σL:
ZR = 2πσ2L/λ
2 ionization effects: refraction due to radial density gradients
3 relativistic self-focusing. Power threshold:
Pc ' 17(ω0
ωp
)2
GW, (26)
4 ponderomotive channelling
All nonlinear effects important for PL > 2TW
Interaction with Underdense Plasmas EM wave propagation Nonlinear refraction 36 133
Relativistic self-focussing: Geometric opticsConsider laser beam with a radial profile
a(r) = a0 exp(−r2/2σ2L),
spot size σ0 just inside a region of uniform, underdense plasma,see Fig. 7.
plasma
θ α
α
∆L
σ
R
laser
a) b)
ZZ
Figure 7: a) diffraction, b) self-focusing
Interaction with Underdense Plasmas EM wave propagation Self focussing 37 133
Geometric optics picture: power threshold
Beam spreading due to diffraction will be cancelled byself-focusing effects if θ = α, (Exercise!),
a20
(ωpσL
c
)2≥ 8. (27)
This represents a power threshold, since the laser powerPL ∝ a2
0σ2L . In numbers:
PL > 9(ω
ωp
)2
GW
This is ∼ ×2 too low because we didn’t take beam profile intoaccount.
Interaction with Underdense Plasmas EM wave propagation SF power threshold 38 133
Focussing threshold – practical unitsLitvak, 1970; Max et al.1974, Sprangle et al.1988
Relation between laser power and critical power:
PL =(mωc
e
)2(
cωp
)2 cε0
2
∫ ∞0
2πra2(r)dr
=12
(me
)2c5ε0
(ω
ωp
)2
P
' 0.35(ω
ωp
)2
P GW.
The critical power Pc = 16π thus corresponds to:
Power threshold for relativistic self-focussing
Pc ' 17.5(ω
ωp
)2
GW, (28)
Interaction with Underdense Plasmas EM wave propagation SF power threshold 39 133
Focussing threshold – example
Critical power
Pc ' 17.5(ω
ωp
)2
GW, (29)
Example
λL = 0.8µm, ne = 1019 cm−3
⇒ ne
nc=(ωp
ω
)2=
1019
1.6× 1021 = 6× 10−3
⇒ Pc = 2.6TW
Interaction with Underdense Plasmas EM wave propagation SF power threshold 40 133
Transverse plasma responseCigar-shaped pulse: take ∇ = ∇⊥, and apply Poisson’sequation (φ and n normalized as before)
∇2⊥φ = k2
p (n − 1),
to obtain density perturbation:
n = 1 + k−2p ∇2
⊥γ. (30)
0 1 2 3 4 5kpr
0.0
0.3
0.6
0.9
1.2
1.5
Ele
ctro
nde
nsity
n(r)
a0=2.0a0=1.0a0=0.5a0=0.2
Interaction with Underdense Plasmas EM wave propagation Ponderomotive channel formation 41 133
Cavitation condition
Consider Gaussian pulse profile a(r) = a0 exp(−r2/2σ2). Aftertime-averaging over the laser period, the density depressionterm in cylindrical coordinates can be written:
∇2⊥γ =
14γ∇2⊥a2 =
14γ
4a20
σ2
(r2
σ2 − 1)
exp(−r2/σ2).
The deepest depression is on the laser axis at r = 0, giving
Cavitation condition, (n = 0):
I18λ2µ >
120
n18σ2µ, (31)
– quite easily fulfilled with TW lasers.
Interaction with Underdense Plasmas EM wave propagation Ponderomotive channel formation 42 133
Relativistic beam propagationNumerical solution of NLSE with an initial radial Gaussian beamprofile with σ0 = 7.5 µm and pump strengths a0.Beam powers P/Pc : 0.5, 1.0 and 3.0 respectively.
0 1 2 3 4
z (mm)
0
5
10
15
20
25
30
Bea
mra
dius
r(
m)
vacuumP=Pc/2P=PcP=3Pc
Interaction with Underdense Plasmas EM wave propagation Ponderomotive channel formation 43 133
Plasma (Langmuir) wave propagation
Without the laser driving term (Ay = 0), Eq.(23) describes linearplasma oscillations with solutions
Ex = Ex0 sin(ωt),
giving the dispersion relation:
−ω2 + ω2p = 0. (32)
The linear eigenmode of a plasma has ω = ωp.
Including finite temperature Te > 0 yields the Bohm-Grossrelation:
ω2 = ω2p + 3v2
t k2. (33)
Interaction with Underdense Plasmas Plasma wave propagation Dispersion 44 133
Numerical solutions – linear Langmuir wave
Numerical integration of the electrostatic wave equation on slide1 for vmax/c = 0.2
0.0 3.1 6.2
p
-0.5
0.0
0.5
1.0
1.5
u,E
,ne/
n 0
uEne/n0
NB: electric field and density 90o out of phase
Interaction with Underdense Plasmas Plasma wave propagation Numerical solutions 45 133
Numerical solutions – nonlinear Langmuir waves
Parameters here: a) phase velocity vp/c ' 1 and vm/c = 0.9; b)vp/c = 0.6, vm/c = 0.55
0 2 4 6 8
p
-2
0
2
4
6
8
u,E
,ne/
n 0
a)
0.0 3.1 6.2
p
-2
0
2
4
6
8
u,E
,ne/
n 0
b)uEne/n0
Typical features : i) sawtooth electric field; ii) spiked density; iii)lengthening of the oscillation period by factor γ.
Interaction with Underdense Plasmas Plasma wave propagation Numerical solutions 46 133
Maximum field amplitude - wave-breaking limitFor relativistic phase velocities, find
Emax ∼ mωpc/e
– wave-breaking limit – Dawson (1962), Katsouleas (1988).
Example
me = 9.1× 10−28g
c = 3× 1010cms−1
ωp = 5.6× 104(ne/cm−3)1/2
e = 4.8× 10−10statcoulomb
Ep ∼ 4× 108(
ne
1018 cm−3
)1/2
V m−1
Interaction with Underdense Plasmas Plasma wave propagation Wave breaking 47 133
Wakefield excitation
fpond
+
+
+
t1
fpond
eEx
t2 +
+
+
eEx
t3
fpond
+
+
+
Interaction with Underdense Plasmas Wakefield excitation 48 133
Resonance conditionThe amplitude of the longitudinal oscillation will be enhanced ifthe pulse length is roughly matched to the plasma period:
τL ' ω−1p .
Example
What plasma density do we need to match a 100 fs pulse?
ωp ' 5× 104n1/2e s−1
Matching condition:
ne ' 4× 1014τ−2ps cm−3
For 100 fs, need ne = 4× 1016 cm−3.
Interaction with Underdense Plasmas Wakefield excitation 49 133
Numerical solution: small laser amplitude
0 5 10 15 20 25
kp
-0.02
0.0
0.02
0.04
0.06
wak
efiel
d
Enpump
Interaction with Underdense Plasmas Wakefield excitation 50 133
Numerical solution: resonance condition (smallamplitudes)
0 5 10 15 20kp L
0.0
0.001
0.002
0.003
0.004
0.005
Em
ax
Interaction with Underdense Plasmas Wakefield excitation 51 133
Numerical solution: large laser amplitude
0 2 4 6 8 10 12 14 16 18 20
kp
-1
0
1
2
3
wak
efiel
d
pump
En
Interaction with Underdense Plasmas Wakefield excitation 52 133
Wake amplitude scaling in nonlinear regimeMurusidze & Berzhiani, 1990
Analytical solution possible for a square pump in the limit βg → 1⇒ Scaling of the wake-variable maxima:
φmax ∼ γ2⊥ − 1
Emax ∼γ2⊥ − 1γ⊥
(34)
pmax ∼ (γu)max =γ4⊥ − 12γ2⊥
where γ⊥ = (1 + a2)1/2 as on p. ??.
Interaction with Underdense Plasmas Wakefield excitation 53 133
2D wakefield excitation
Interaction with Underdense Plasmas Wakefield excitation 54 133
Electron acceleration by wakefields
Conventional synchrotrons and LINACS operate with fieldgradients limited to around 100 MVm−1.
Plasma is already ionized; can theoretically sustain a field 104
times larger, given by:
Ep =mecωp
eε
' n1/218 ε GV cm−1, (35)
where n18 is the electron density in units of 1018 cm−3.
Interaction with Underdense Plasmas Wakefield excitation Electron acceleration 55 133
Laser-electron acceleratorTajima & Dawson, 1979
Laser-driven wakefields must propagate with velocitiesapproaching the speed of light (vp = vg < c).Plasma wave has a phase velocity:
vp = c
(1−
ω2p
ω2o
) 12
' c(
1− 12γ2
p
), (36)
where γp = ω20/ω
2p .
Interaction with Underdense Plasmas Wakefield excitation Electron acceleration 56 133
Acceleration length
A relativistic electron (v ' c) trapped in such a wave will beaccelerated over at most half a wavelength in the wave-frame,after which it starts to be decelerated.Effective acceleration length:
La =λpc
2(c − vp)' λpγ
2p
=ω2
ω2pλp
' 3.2 n−3/218 λ−2
µm cm. (37)
Interaction with Underdense Plasmas Wakefield excitation Electron acceleration 57 133
Maximum energy gain
Combine Eq. (35) and Eq. (37) to obtain the maximum energygain:
∆U = eEp.La
= e(mωpc
e
)εω2
ω2p
2πcωp
= 2π(ω
ωp
)2
εmc2
' 3.2 n−118 λ
−2µm GeV. (38)
Interaction with Underdense Plasmas Wakefield excitation Electron acceleration 58 133
Limiting factors
In principle, TW laser is capable of accelerating an electron to5 GeV in a distance of 5 cm through a plasma with density1018 cm−3.Spoiling factors:
Diffraction: typically have ZR � La, so some means ofguiding the laser beam over the dephasing length is essential
Propagation instabilities – beam break-up: modulation;hosing; Raman
Interaction with Underdense Plasmas Wakefield excitation Electron acceleration 59 133
Bench-Top Particle Accelerators
Standard electron acceleration in fast plasma wave (recap):
Acceleration length
La ' 3.2 n−3/218 λ−2
µm cm
Energy gain∆U ' 3.2 n−1
18 λ−2µm GeV
Interaction with Underdense Plasmas Wakefield excitation Bench-Top Particle Accelerators 60 133
Acceleration mechanisms
Large variety of attributed acceleration mechanisms inexperiments (long and short pulse):
self-modulated
forced-wave
wave-breaking
guided
bubble-regime
GeV milestone reached September 2006 (LBL). More to comeon Wed–Fri ....
Interaction with Underdense Plasmas Wakefield excitation Bench-Top Particle Accelerators 61 133
What mechanisms are at work here?
Interaction with Underdense Plasmas Wakefield excitation Bench-Top Particle Accelerators 62 133
Livingstone chart for laser-plasma electronaccelerators
1980 1985 1990 1995 2000 2005 2010
Year
1
2
5
10
2
5
102
2
5
103
2
Max
.ele
ctro
nen
ergy
(MeV
)
IC/RALUCLA
INRS
ILE
UCLA
LULI
RAL
ILECUOS
NRL
RAL
CUOS
MPQ
LOA
RAL
RAL (Astra)
LBL
LOA
LBL
Figure 8: Open circles represent early beat-wave experiments; filledcircles single pulse wakefield experiments; trianglesquasi-monoenergetic electron beams
Interaction with Underdense Plasmas Wakefield excitation Bench-Top Particle Accelerators 63 133
Summary of Lecture 2
Plasma response
Cold fluid equations
EM wave propagation
Plasma wave propagation
Wakefield excitation
Interaction with Underdense Plasmas Wakefield excitation Bench-Top Particle Accelerators 64 133
Mitg
lied
derH
elm
holtz
-Gem
eins
chaf
t
Physics of High IntensityLaser Plasma InteractionsPart III: Interaction with Solids
20–25 June 2011 Paul Gibbon
Lecture 3: Interaction with SolidsShort pulse interaction scenarios
Collisional AbsorptionNormal skin effect
Collisionless AbsorptionResonance absorptionBrunel model
Hot Electron GenerationScalingMeasurements of hot electron temperature
Ion accelerationMechanismsSheath modelHole boringLight sail
Interaction with Solids 66 133
Short pulse vs. long pulse interactions
Long-pulse interaction physics (ICF – ns lasers):
Collisional heating and creation of long scale-length plasmas
Laser reflected at critical density surface
Fast (keV) particles produced at ’high’ intensities(1016 Wcm−2)
Femtosecond pulses
Pulse length typically < ion motion timescale(hydrodynamics)
Huge intensity range 107
No single interaction model possible
Interaction with Solids Short pulse interaction scenarios 67 133
Typical interaction scenario: ICreation of critical surface
Comination of field and collisional ionization over the first fewlaser cycles rapidly creates a surface plasma layer with a densitymany times the critical density nc .
ω2 =4πe2nc
m, (39)
where e and m are the electron charge and mass respectively.
In practical units (see Eq. 15):
nc ' 1.1× 1021(
λ
µm
)cm−3. (40)
Interaction with Solids Short pulse interaction scenarios 68 133
Interaction scenario: IIIonization degree
Example
Al foil target Al has 3 valence electrons; 6 more can be releasedfor a few hundred eV. The electron density is given by:
ne = Z ∗ni =Z ∗NAρ
A. (41)
effective ion charge: Z ∗ = 9atomic number: A = 26Avogadro number: NA = 6.02× 1023
mass density: ρ = ρsolid = 1.9 g cm−3
electron density: ne = 4× 1023 cm−3
density contrast (1 µm): ne/nc ' 400
Interaction with Solids Short pulse interaction scenarios 69 133
Interaction scenario: IIIHeating
Target is heated via electron-ion collisions to 10s or 100s of eVdepending on the laser intensity.The plasma pressure created during heating causes ion blow-off(ablation) at the sound speed:
cs =
(Z ∗kBTe
mi
)1/2
' 3.1× 107(
Te
keV
)1/2(Z ∗
A
)1/2
cm s−1, (42)
where kB is the Boltzmann constant, Te the electron temperatureand mi the ion mass.
Interaction with Solids Short pulse interaction scenarios 70 133
Interaction scenario: IVExpansion
Because of ion ablation, density profile formed is exponentialwith scale-length:
L = csτL
' 3(
Te
keV
)1/2(Z ∗
A
)1/2
τfsÅ. (43)
Example
100 fs Ti:sapphire pulse on Al foil heats the target to a fewhundred eV→ plasma with scale-length L/λ= 0.01–0.1. (cf:100-1000 for ICF plasmas).
Interaction with Solids Short pulse interaction scenarios 71 133
Collisional absorption
Standard method for electromagnetic wave propagation in aninhomogeneous plasma – see books by Ginzburg, Kruer.
Start from Maxwell’s equations with small field amplitudes and anon-relativistic fluid response including collisional damping:
m∂v∂t
= −e(E +vc×B)−mνeiv , (44)
where νei is the electron-ion collision frequency.
Physically arises from binary collisions, resulting in a frictionaldrag on the electron motion.
Interaction with Solids Collisional Absorption 72 133
Electron-ion collisional frequencySpitzer-Härm
Collision rate:
νei =4(2π)1/2
3neZe4
m2v3te
ln Λ
' 2.91× 10−6ZneT−3/2e ln Λ s−1. (45)
Z = number of free electrons per atomne = electron density in cm−3
Te = temperature in eVln Λ is the Coulomb logarithm, with usual limits, bmin and bmax,of the electron-ion scattering cross-section.
Interaction with Solids Collisional Absorption 73 133
Coulomb logarithm
Limits are determined by the classical distance of closestapproach and the Debye length respectively, so that:
Λ =bmax
bmin
= λD.kBTe
Ze2 =9ND
Z, (46)
where
λD =
(kBTe
4πnee2
)1/2
=vte
ωp, (47)
andND =
4π3λ3
Dne
is the number of particles in a Debye sphere.
Interaction with Solids Collisional Absorption 74 133
Absorption in steep density profiles: skin effect
Density profile can be approximated by a Heaviside stepfunction:
n0(x) = n0Θ(x),
giving a dielectric constant (cf: dispersion relation Eq. 24):
ε(x) ≡ c2k2
ω2 = 1−ω2
p
ω2(1 + iνei/ω)Θ(x). (48)
Interaction with Solids Collisional Absorption Normal skin effect 75 133
Solution for laser field I
For normally incident light, the transverse electric field has thesolution
Ez =
{2E0 sin(kx cos θ + φ), x < 0E(0) exp(−x/ls), x ≥ 0
(49)
where ls ' c/ωp is the collisionless skin-depth, k = ω/c, E0 isthe amplitude of the laser field and φ a phase factor.
Matching vacuum and solid solutions at the boundary x = 0gives:
E(0) = 2E0ω
ωpcos θ
tanφ = −lsω
ccos θ.
Interaction with Solids Collisional Absorption Normal skin effect 76 133
Solution for laser field I
-4 -3 -2 -1 0 1 2kx
0
1
2
3
4
5
6
Fie
ldin
tens
ityEz
2By
2
E2(0) = 2E02nc/n0
n0/nc
Interaction with Solids Collisional Absorption Normal skin effect 77 133
Reflectivity: Drude model
Example
Al: Z ∗ = 3, ne ' 2× 1023 cm−3, λL = 0.8 µm , ne/nc ' 100.
0 15 30 45 60 75 90
(o)
0.0
0.2
0.4
0.6
0.8
1.0
a=
1-R
FresnelHelmholtz
Figure 9: Angular absorption for a step-profile with ne/nc = 100 andν/ω = 5 calculated analytically from the Fresnel equations (solid) andnumerically from the Helmholtz wave equations (dotted).
Interaction with Solids Collisional Absorption Normal skin effect 78 133
Collisional frequency turn-offquiver velocity correction
Effective collision frequency reduced by quiver motion in laserfield
νe� ' νeiv3
te
(v2os + v2
te)3/2. (50)
A temperature of 1 keV corresponds to a thermal velocityvte' 0.05, so collisional absorption starts to turn off forirradiances Iλ2≥ 1015 Wcm−2µm2.
Interaction with Solids Collisionless Absorption 79 133
Collisionless absorption mechanisms
What other absorption mechanisms can couple laser energy to ahot, solid-density target?
1 Resonance absorption (L/λL � vos/ω) – Denisov (1957)
2 Anomalous (collisionless) skin effect – Weibel (1967)
3 ’Vacuum heating’ – Brunel (1987), Gibbon (1992)
4 Relativistic j × B heating – Kruer (1988), Wilks (1992)
All of these mechanisms will generate fast electrons withenergies Th ∼ keV–MeV.
Interaction with Solids Collisionless Absorption 80 133
Collisionless resonance absorption
nc
n cosc2
x
θ
en (x)
yθ
B
E p
kE
Bs
Figure 10: Standard picture of resonance absorption: a p-polarizedlight wave tunnels through to the critical surface (ne = nc) and drivesup a plasma wave. This is damped by particle trapping and wavebreaking at high intensities.
Interaction with Solids Collisionless Absorption Resonance absorption 81 133
Resonance absorption: Denisov function
Self-similar dependence on the parameter ξ = (kL)1/3 sin θ,kL� 1.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
=(kL)1/3 sin
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
()
Interaction with Solids Collisionless Absorption Resonance absorption 82 133
Absorption rate for long scalelengths
To a good approximation,
φ(ξ) ' 2.3ξ exp(−2ξ3/3), (51)
and the fractional absorption is given by:
ηra =12
Φ2(ξ).
Behavior nearly independent of the damping mechanism.
Interaction with Solids Collisionless Absorption Resonance absorption 83 133
Kinetic simulation of resonance absorptionParticle-in-Cell
0 2 4 6 8 10x/
0.0
0.5
1.0
1.5
2.0
x
n e/n
c
a)
0 2 4 6 8 10x/
0.0
0.05
0.1
0.15
Ey
b)
0 2 4 6 8 10x/
0.0
0.05
0.1
0.15
Bz
d)
0 2 4 6 8 10x/
-0.2
0.0
0.2
0.4
x
...
..
.. ... ...
.. . ... .. .
..
..
. ... ...
. .. ...
.
.... . .
... . .
... ..
..
.
. ...
... ..
... ..
...
.
...
.. ..
.... ..
. ... ... ... .
..
. ...
..
.. ..
. ... .. .....
.. ...... ....
.. ....
....
... .
.. .
...
..
.... .
... .....
.
.
.. ... ...
...
.........
. ..
... ..
..
.
... ..
..
. .. . ........
.
..
.
. ..
.. . .
....
... .
.. ...
. ..
. ..
....
...
...
.. . ... .
..
..
....
..
. .. .
..
..
..
..
.. ....
. ... ...
...
..
.
.
. ..
. .
..
. . ..
.... ...
.... .
.. .
...
.
.
... ..
.
... .
...
.. ...
..
.
.
. ....
. ....
.. ....
........
..
..
..
.... ..
..
.. .
.
..
.
. ......
..
..
.
..
... .
. .
.. .. .... ... ..
...
. ....
.. ..
.
.
... ...
.. .... . .
.
...
..
.
. ....
. .... ... ....
..
...
..
.. .
.. .. ....
... .. .
..... .
......
.. ..
.. ... ... .
..
.
.
...
.
..
. ...
..
.. .
...
. ......
..
..
. . ...
... .
.. .. .
..
.
.. .
..... .
..
.... ..... .
.. ..
.
..
.
..
..
.
... .
....
..
..
.
.....
.....
. .. ...
.. .....
.. .
.
. ......
.
..... .
. ..
. ....
.
.
..
.
... .
..
. . ..
.
.... ..
..
.
. ........
..
..
.
. ....
. .. .
..
..
.. ..
. .. ...
..
. ..
.
..
.
... .
....
..
...
....
.
.
.... .
..
. ... .
... .
.
...
.. ..
. .
....
..
. ..
...
.
...
..
...
.. ..
...
...
..... ...
..
.
.. .
... .
....... . ..
.. ....
.
. ..
..
.. .
.
..
..
.... ..
... ....
... . ....
..
..
... .
....
..
...
...
...
...
.... . ...... .....
.... ..
.
.
.. ..
..
..
.
.
.
.
. .
.. ....
. ....
.
.. .
.. .
.
.. . .... ...
...
.... .. ..
..
...... .. .
..
.
.
..
.
. ....
. . . .. ... .
. . ...
..
..
...
... . .. .
. .
.. .. .
. . .
..
.
.
...
.. .. .. . .
. .
. ...
...
..
..
.
.. .
..
.. .
... .
....
.. .... .
.
.
.
.....
.....
... .. .
... .
..
.. .
.
.....
....... . ..
.....
.. .. .....
....
..
.. ... . ..
.. .. .. .. .. .
...
... ...
..
....
...
.
...
.
... . .. ...
.....
...
.. .
... .
. ....
.
..
..
...
..
..
....
. ..... . .
..
.
.
..
.. .....
...... .
. .
.
. ...
... ...
. ...
....
..
..
.. .
.
... .. .. .
. .. .
....
..
..
. .....
..
. .. .
...
.
.. ..
. .. .. ..
. ....
.
.
.
...
. .....
..
.
.
..
. ....
. ...
...
..
. ..
.. ... ...
. ..
.. .... . .
..
.
.
.
.. .. ..
... .
.. .. .. ..
.. ..
. ....
.
.. .. ..
.. .
..
.. .....
.
..
.
...
..
... ..
..
.. ..... ...
.. .
...
..
. ... .
... .
...
.
.
..
.. .
.. ..
...
.
. ...
....
.
.
...
..
.... ......
..
.
...... ...
...
...
..
. ...
.. . .
..
. ....... ..
... .. .
.
. ...
....
.
.... ..
... ...
...
...
...
..
. . .
. ..
...
. . .. ..
.....
.
. . ...
.
...
.
..
.. . ..
. ..
... . ...
.
....
.. .
... .
... .. .
.
..
.
...
. ...
.
...
...
....
...
..
..
.
.
...
.. ..
.
.... ..
.. .. .
.
. ...
...
.. .
.. .
.
.....
....
.. ..
..
.... . ...
...
..
.
..
..
.. ..
...
...
..
..
. ...
.
.
...
...
..
.. . ...
... .. .. .. .
... .
.
..
..
.
.....
.
.
.
..
.. ... ..
.
. .. ..
.. .... .
..
.. .
...
.........
..
.
.
.
..
. .. .
.. ..
.. .
.. ..
.....
... .
... .. ..
. . ... ..
. ..
...
....
.
. ... .
..
..
.
. . .. ..
... ..
.
...
. ..
. .
.
.. . ..
.. .
. ...
.... ...
. ..... .
.. .
..
..
..
. ..
. ...
.
...
..
...
..
.. .
...
...
..
.. .
.... .. .
.... . . ..
..
.. ...
.
. ..
... .... .
...
...
. ... .... ..
..
.
.
...
.. .
..
.
.
.. .
. ..
..
.
. ... .
....
....
....
.
. ... ...
..
.. .
..
.. ...
.. ..
...
...
...
...
.
..
..
..
.
..
...
..
.
.. . . .
.
.. .
...
.... ... .
..
.....
...
.... ...
. ....
..
..
.
.
...
.. ...
...
.. .. .
..
..
.
. . ....
..
..
...
..
..
....
. ..
.. .. .
....
.. .....
..
.
...
.
.. .
...
.. ...
... ..........
. . ..
..
..
... ..
.
.. ...
...
.
.
... .
...
...
... .
.
..
. ....
...
....
.
.
.. . .
..
...
...
.
...
...
...
....
... .
..
...
...
.
..... .
..
. ..
..
....
.
..
. ....
..
..
..
....
.
..
...
. ....... ......
.
...
...
..
... ..
....
...
...
.
.
..
.... .
. . ..
.
. ......
... ..
..
....
.. .
.
..
...
...
.. .. .. ..
.
...
..
.. ....
... ..
. ..
.
.
...
..
. ...
. ...
. .... ..
.
... ..
.. . ..
...
....
.... .
.
. ..
...
..
... .. ..
... ..
. .. .
...
..... . . ...
.. ....
... ...... ..
.. .....
. . ..
... .
..
...
.. .
...
.
...
..
.. ..
.. ..
..
..... . . .
. .....
...
. ...
....
. ...
....
....
.
..
..
. ... .
. .
.. ...
...
...
.
.
... .
... .
..
.. . ..
... . ..
. ..
...
.
.
...
.
.
.
. ...
...
.
.. . .
..
..
. ..
.. ..
.
..
... .
. .. .... .
.
.
.
. ... ..
...
..
...
..
.. ...
.
.
.. ..
..
. ...
.....
.... .
. ...
. ..
.. ..
..
..
..
.
.. ..... ....
.. .
. ....
... ..
...
. ....
. ..
. ...
.
. .. ..
.
.
.. .. ..
...
....
...
...
...
.
.
. ....
... .
.
..
.. ....
.. .
.. .
...
.. .... .....
. ...
..
....
.. . ...
..
..
.. .
.
...
.. ... .. .
... ..
..
. ... ... . ....
.. ..
.
....
.. ..
....
... .. .. ..
...
.
...
..
...
... . ..
...
...
. ..
..
.
... ....
.. ...
... ..
.
. ..
.
. . ..... .... .. ....
.. ..
.
.. ... .
.. ..
. ..
.
. .
.. .
. .
.
. .
... .. .
. ..
. . .
.. . .
.
.
.
.. . ...
. .... .. . ...
..
.
. .. .. ..
. ..
. ...
..
.
. ...
..
.
.... ...
. .....
...
. ... .
. ... .....
..
..
... .. .
. ..
.
.
. ..
..
. ... . ..... ..
. .. .
..
..
.. .
... .
. ...
... .. ... ...
....
.....
....
...
.
...
...
.
.. ...
..
. ..
..
.
... .... ....
. ...
. ...
.
.
.
. ...
. ..
..
. .
..
...
.
... ... . .
...
... .
...
.. .
...
...
.
...
. ..
. ... .. .
... .
.
.. ..
.
..
..
. ...... .
...
..
..
...
. . ...
..
...
.....
.... .
... .
... .
. .. .. ....
.
.
... ... ..
..
....
...
.. .
... ..
..
..
.
....
.
.
....
....
...
..
.....
... ..
. .. .. .. ..
..
..
.
....
..
.. .
..
. ..
..
....
..
.. .. ...
. ..
.. . ..
.
.
...
..
..
.
....
.
..
. ..
. ...
..
... .
.
.
.
. .. ....
.
.
...
..
...
.
.
. ...
. ..
.
..
.
..
.
... .
. ...
.... .
..
..
.
... ....
.
..
....
...
. ...
..
. ...
.. ...... .
...
.
..... .
....
.. .. .
..
.
...
....
.
...
.
.
. .
.
.
.
. ..
.
.
.
.
...
.
..
.
. .
..
... . .. ....
...
...
. ..
. ..
.
...
...
..
...
. ..
..
.. .
.
..
.. .
.. ..
.
.
.. .
.. .
..
.
. .. .
. ...
..
.
.
..
.
.
.
.
.
.
..
.
.
....
...
....
..
.. .
.
..
.
.
... ...
.... .
...
... ...
......
.
.
.
. ..
.. .
..
.. ...
...
.
.
.. ..
.. ...
. .....
....
.. .
. ..
.
.
..
.
.. .. .
. ... ..
.
..... .. .. ..
..
.
.. ...
.
...
....
. ..
..
...
... .. .... .
..
..
.
.
. ... .
.
.
.
.
...
... .
.
..
.. .. .. ....
..... .
.. .
.
.
...
...
.
.... .. ..
... ...
.
.. ...... .
..
..
.
..
....
....
..
. ..
..
.
.
..
.
..
.
..
.
..
.
. .
...
.
...
...
.
.
.
....
..
... .. .
..
. .
.
.
..
.... ..
.
...
..
... . ... .
.
. ..
..
.
.. .
..
.
...
.
. .. ..
.
.
... .
..
. .
. .. ...
..
.
.. ....
..
..
.
...
.
..
.
. ..
..
.
..
..
. .. ... .
.
.. .. .
. .
.
......
..
. . ..
..
..
.... ..
..
. ...
.
...
.
... .. .
.
..
.
.. .....
..
....
..
.
..
.
. .. .
..
...
...
.. .
. .
.
.
. ..
... ..
.... ... .
.
.
.
....
.
..
. .. .
...
..
.
...
... ...
.
. ..
...
.. . .
....
...
... .. ...
.. ...
...
.... . ...
. ..
..
..
. ...
..
.. .
... .
....
..
.
... .
.... ..
.
..
..
.
.
.. . .
.
.. .
.... ..
..
.. ..
....
.. ..
..
..
.. ..
..
..
.
..
..
.
... .
.....
.. .
..
.. ..
.
... .
..
....... .
..
.
.. .
.....
.
.. .
.
...
.
.
...
...
.
..
... .
. .
.... .
.
.
.
... .
..
..
...
..
... ....
..
..
..
.
.. . .. ..
. ... .. .
..
... . .... .
.
... .. ..
.
.....
...
.
..
..
.
......
... . ..
.. ... .
.. .. .
..
. .
.
.
.. .
...
.
.
.
..
.
.
.. .
. .. .
.
... ..
.
...
.
..
..
.... .
.
..
.. .
..
.... ..
.
.
.
..
... .. .
.
... .
.
.
.. .....
... ..
.
....
.
... .. ...
.... .. ..
...
.. ... ..
..
.. .
.
. .
.
... ..
. . ..
.
.
.
.
.
..
. ..
...
... ......
..
.
...
..
..
.
... ...
.....
.
..
...
..
.
....
.
. .. .
..... ..
.. ....
..
. .
.
....... ...
..
.
. ... ..
.. ... .....
.
.
...
..
..
..
.
..
...
.. .....
.
....
...
..
.
.
. .....
.
..
. ....
.....
. ....
.... ..
.
... .
.. ....
.
. ..
.. ...
.
.
....
...
.. .
.
...
.
.. .. .
.
.
.
..
..
..
.. ....
.
.
.
.
.
.
..
....
.. ...
.
..
.
... . .
..
. ..
.
. ..
.
.
.. ... ..
. .
.
. ..
...
. .
... ....
. ..
..
.
.
.
. ..
.
.
.. . .
.
...
. ....
...... .
.. .
....
.
.
.
.
.
. . ...... ..
.
.
....
. . ...
..
. .
.
.. ...
.
.. ..... .... .
.
... .
.
....
.
..
..
.
.
...
.
...
.. .
.
.
.
.
.
.
.. . .
.
...
...
.
.. ...
...
.
. ..
.
.
....
.
.. . ...
.
..
. .. ....
..
...
.. .
. .
.......
..
..
.. . ..
.. .
.
.
.. .
..
. ...
.
..
..
.. .
.
..
...
.
..
. ...
.. .... .
. .
...
...... ..
.
.
..
.. .. ..
...
.
. .
.
.
..
... .
...
.... .
.
. ..
.
.
. ...
...
...
.
.
.
.
.
.. .
.
.
...
.. .
..
..
. ... ... .
.
.
...
.
..
. ...
.
.
.....
. .
.
... .
.
..
...
.. .
..
..
.
... ...
.
.
..
.
....
.
.
.. .
.. ... .
.
..
.
.
..
...
.
.. ..
.
..
..
.
...
.
.. .. ..
....
.. .....
...
.... .. ..
... .....
.
..
.. ......
.
.
...
.. ..
.
.
.
..
...
....
.
..
...
. .... ..
.
.... .
.
.. ..
.
....
.
..
..
.
.. .
... .
... ...
..
.
...
...
..
...
...
.
. .. ... ....
.
.
.
.
.. .
. .
.. ... . .
..
.... ..
.
..
. . ...
.
.
..
..
..
.
.
.
.
.
..
.
.
..
.
..... .. .
.. .
...
..
.
...
.
...
.. .
...
. ..
. .
..
.
..
. . ...
.
.. ...
. .. .. .
.
.
..
.....
....
.
..
..
.
..
..
. ..
.
. . .. .. ..
.
..
.. ... .
.
.
.. ...
.
..
...
.
.
.
. .
. ...
..
..
..
.
..
...
...
.
.
.
..
..
.
... .
..
..
....
.
.
..
.... ..
.
..
.
.
..
.
..
. ...
...
.
.
..
...
.
.
. ..
.
.
..
. ..
..
..
..
..
..
....
..
. .. .
.
.
.. .. .
.
..
.
. ..
.
. ..
..
...
.
.
.... .
.
..
.. ...
...
..
..
.
.
.
..
..
.
...
.
.
..
. .
...
.
.
...
.. . ..
. .. ..
......
. ..
.
.....
.
..
..
.. ..
.
.
...
.... .
.
.
... .
.
..
.
.
.
....
.
...
..
.
.. ...
.
.
..
.
.
.
. ..
.
. ...
. .
..
..
.
. .
. .
.
....
.
..
.
. ..
...
. ....
....
... .
.
.
..
.
.
...
.. .
.
..
..
.. ..
....
.
.
.
.
....
.
.
..
... .
. .. ..
. ... ... .
....
.
.
.
.
.. . ...
...
...
.
.
. .
.
.
..
.
...
..
..
.
.
...
..
....
..
. ..
.
.
...
....
.. .
.. .
....
.
..
.
.
..
..
. .
..
...
.
.
..
.
.
..
.
.
.
...
. ..
.. ..
.
....
..
.
....
....
..
... .
..
.
..
..
. . ..
.
..
..
.....
.
..
.
.
..
.
...
...
.
..
...
.
..
.
..
.
.
..
..
.
.. .. ....
..
.
.
.
.
..
.
.. .
.
..
.
. .
.
..
.
.
.. .
.. .
.
.
.....
.. .
..
.
.
.
.
.
.. ..
.... .
.. .
.
... .. ..
.
.
...
... . ..
..
.
..
.
.
.
..
..
....
.
.
.. .
.
.
.
.
.
.
..
.
.
.. ...
.
...
..
.
.
..
.
....
. .. .
.
.
..
....
. ..
.
..
.
.
.
.
.
.
. ... .
.
...
..
.. .. .
.
.
.
..
.
...
.
. ..
.
... .
.
..
.
.
.
.
.
..
.........
..
...
.
..
.
.
.
.... .
.
..
..
.
.
..
. ...
.
.
.
..
.
...
.
.
.
..
..
....
.
.
..
.
.. .
.
. .....
.
...
. .
.
..
...
..
.
. ...
..
.
.
.
..
.
... .
..
.
..
.
.
...
.
.. .
.
.
..
.
..
.
.
..
.
.
... ..
.
....
.
.
..
...
...
.....
.
... .
. ..
.
..
.
.
.
.
.
.
..
. ....
.
... .
..
..
..
.
.
.
.
.
.
...
.
.
.
.
.
.
.
.
.
.
...
...
.
.
.
.
..
..
.
. .. . .
.
...
.
..
.
..
.
..
.
.
. .
.
..
.
.
. ..
..
.
..
.
...
..
....
....
.
.. .. .
.
.
..
.
..... . ...
.
.
..
.
...
..
.
.
..
.
.
.
.
.
...
....
.
..
..
.
..
.. . ..
.
..
..
..
....
. .
..
.
.
. .
.
.
.
.
.
...
.
.. .
.
.
..
.. ..
.
.
.
.
.
. .
.
..
.. .
.
....
.
..
...
..
.
....
.
.
...
....
.
...
.
..
.
.
..
.
..
.
.
.
.. . .
. ...
.
.
.
.
.
. .
.
..
..
...
....
.
.
. .
. ... ..
.
.
.
..
.
.
. .. ....
.
..
...
.
..
.
.
.
... .
.
.
.. .
.
.
.
.
.
..
...
.
.
..
... ..
.
.
.. .. .
.
. ...
.
....
.
..
.
..
..
....
.
.
....
...
. ..
..
. .. ......
.. .
.
...
.. ..
.
.
.
...
..
.
...
.
..
.
.
...
.
.
.
...
.
...
...
.
..
.
..
. ..
... ....
.
.
....
.. ....
..
.
...
.
.
.
. ..
.
..
. ..
.
.
..
.
.
.
. .
.
.
.. .. .
..
.
... ..
.
..
.
..
.. .
.
...
..
...
.
.
.
.
..
.
.
.
.
.
.
..
.
.
.
.. .
. . .
.
.
.
...
.
.
.
.
..
.
...
.
.
.
.
...
..
. ...
.
.
...
.
..
.
.
.
.
...
.
..
.. ..
.
..
. . .
.
.
..
.
..
.
...
.
.
.
.
. ..
..
.
..
..
.
..
.. .
.
..
.
..
.
..
.. .
.
...
.
.
.
.
.
..
. .
.
.
... ...
.
.. . . .
..
....
.
.
.
..
.
. .
. ...
.
..
.
...
..
.
..
..
. .
.
.
.
..
.
.
.
..
.
.
.
.... .
.
.
.
...
.
.
..
.
. .
.
.
.
.
. ..
..
..
.
.
.....
.
..
..
.
.
..
..
.
.
.
..
.. .
....
.
.. .
.
..
. .
.
..
. ...
.
.
..
.
..
. ... .
.
...
.
.
.. .
. .
.
..
.
..
.
.
.
..
..
.
.
..
..
.
.
.
.
.
.. .
.
...
.
.
. ..
..
.
.
..
. .. .
.
.. ....
.
.. .
.
.
... ..
... .
.
.
..
.
..
..
...
...
.. ....
...
. .
.
.
..
.. . ..
.
.
.
.
.
...
..
..
.
.
..
.
.
.
..
.
....
. .
..
.
.
.
.
...
.
.
. .
..
..
...
...
..
. ..
.
..
.
.
..
.....
.
.
.
.
.
.
..
...
.
.
.
..
...
.
..
..
.
...
.
.
.
..
..
.
.
.. .
.
.
. ..
.
...
.. ..
...
.
.
.
.
.
.
.
.
.. .
.
.
...
.
.....
.
.
. . ...
...
. .
.
.
..
.. .
..
.
.
.. ..
.
.
.
.
.
. .
...
.
.
.
.
...
..
.
.
.
.
.
.
.
. .
...
..
..
..
.
.
.
..
.
.
.
..
.
..
.
....
.
...
.
.
.
..
..
..
.
.
.
.
.
.
.
..
.
.
.
.
.. .. .
.
...
.
.
.
..
.
.
.
. .
.
.
...
..
.
.
.
..
. ...
..
.
.
...
.
..
....
...
.
.. .
..
.
.
.
. .
.
.
.....
.
...
.
.
.
... .... .. ..
..
.. .
.
.
... .
.
..
.
.
.
..
.
.. ..
.
.
.
..
.
...
.
..
... .
.
.....
.
..
.
.
.
.
. .
.
.
.
.
.
.
... .
.
.
.
.
..
.
.. .
..
.
..
.
..
.
.
.
..
..
..
.
..
.
.
..
. ..
...
.
... .
..
.
.
. ..
.
... .
..
.. ..
...
..
.
. ...
..
.
....
.. .. .
.
.
.
...
...
..
.
.
.
.
..
..
.
.
.
. ..
.
. ...
.
.
.
.
.
....
..
... .
...
.
..
.
.
.
.
...
..
..
. ..
..
..
.
.
.
.
..
..
. .
..
..
.
.
.
.
.
.
..
.
.
..
.
.
.
.
.
. .. .
.
...
...
.
.. ...
. .... ..
. .
.
.
.. ....
.
.
...
.
.
.
.
.. .
.
.
..
..
.
...
...
.
.
..
.. .
...
..
.
.
.
.
..
.. .
.
.
.
.
....
..
. ..
.
. .
.
..
..
..
.
.
.
.
.
.
.
...
.
.
.. .
.
.
.
..
..
..
.
.
.
...
...
.
..
.
...
. ..
.
.
.
.
...
.
..
.
.
...
..
.
...
.
.
... .
..
..
..
.
.
..
.. .
. . .
..
.
.. .. .
. ..
..
...
...
. ..
.
.
.
..
.
...
..
.
.
....
.
.
.
. .
..
.
.
..
.. .
.
...
.
..
.
.
.
.
.
.
..
.
. .
.
..
.
...
..
..
.
.
..
.
..
.
. .
.
..
.
.
.
..
.
.
.
.
.
.. . ..
.
.
..
.
.. .
...
.
.
.
..
..
.
... ..
.
..
.
.
.
.
.
..
.
.
.
.
..
..
.
..
...
..
.
..
..
.
.
.
...
.
..
....
.
.
.
..
.. ..
.. .
..
..
.
.
.
..
.
.
.. .
..
.
. ...
...
.
.
.
.
.
.
..
.
.
.
..
. ..
. .. .
.
..
. .
..
..
. ..
.
. .
.
.
..
..
..
.
. ...
..
. ..
.
.... .
.. ..
.
.
.
..
.
.
..
..
.
..
.
....
. ..
.
..
.
.
.
.
.
..
..
..
..
.
.
.
. ..
..
. .
.
..
.
.
. ..
..
.
.
.
.
..
..
...
...
..
.
..
...
..
..
.
.
.
..
..
.
...
. .
.
..
... ..
....
.
.
..
...
.
..
...
.
..
..
.
..
..
..
.
.
.
.
..
.
.
.
...
.
.. .
..
.
.
.
.
.
.
. ..
.
.
.
...
..
. ..
.
.....
. .
. ..
..
.
.
...
.
..
.
. .
.
..
..
.
..
.
..
. .
.
...
.
. ... . ..
..
.
.
.
.
..
.
.
.
.
. .. .. ..
.
....
.
. . .
.
. . ...
..
...
...
.
..
.
.
. .
.
...
...
.
.
..
.
. .. .
.
...
. .
.
.
.
.
..
..
..
.
.
.. ...
...
.
...
.
.
. .
.
..
..
.
.
..
.
. .
.
..
..
.
.
.. ..
.
..
.
..
.
.
.. .
.
.
.. .
...
.
..
.
.
.
.
.. .
..
.
.
.
.
.. .
.
..
.
..
. .
.
...
.
.
.
.
.
..
.
.. ...
.
. .
.
. .....
..
..
.
.
.
..
..
.
.
.
..
.
.
..
.....
..
.
.
..
.
.
..
.
...
...
.
.
..
. ..
.
.. .
.
.. . .
..
. ..
.
..
..
.
.
.
.
...
.
.
.
..
..
..
.
..
.
.
.
.
.
.
.
. ..
.
...
.
...
. .
..
.
....
..
....
.
.
..
.
...
.
.
.
. .
.... .
.
...
.
..
.
.
.
.
.
. ...
..
..
..
....
.
.
..
. .
. .. .
..
.
.
..
.
.
. .. ... ..... .
.
...
.
....
...
..
.
.
...
.
.
.
.
. ...
.
.
...
.
..
.. ..
.. ..
.
.
....
.
.
. ..
.
.
.
.
.
..
....
.
.
..
...
.
.
..
.
.
. ..
.
...
..... .. .
. . .
..
.
..
..
.
. .
.
..
.
.
..
.
.
.
...
.
..
..
.
...
..
..
.
..
.
..
..
..
..
..
..
.
.
... .
. ...
.
.
..
.
.
.
.
. .....
.
.
. ..
.
.
.
..
.
...
....
.
.
..
.
. ...
.
. ... ....
.
..
.. .
.
. ... .
.
.
.
.
. ... ..
.
.
..
.
. .
..
.. ..
..
.
..
.
.
.
.
.
.
..
..
.
.
..
.
..
...
.
..
..
..
.
.
..
... ..
... .
..
.
.
.... ....
...
.
..
.
.
.. .
.
. .
.
. ..
.
...
... ..
.
......
.
..
...
..
.
.......
.
.
.
.
.. .
..
... .. .
..
..
.
..
..
..
.
..
..
.
.
. .
..
...
..
.
..
.
.
.. . .
. ... ...
.
..
..
.
. ..
.
.
..
.
..
..
.
.
..
..
.
..
. .
....
..
..
. .
..
..
.
.
..
.
.
.
..
..
.
.
.
.
....
.
.
.
.
..
.
...
.
.
.
.
.
. ... . .
.
.
.. ...
..
.
.
.
. ...
.
..
..
.
.
. .
..
.
.
.
.
.
.
.
..
.
.
.
.
.
...
.. ..
...
..
.
.
.
.
..
.
.
.
.
..
.
.. ...
. .
..
.
.
...
.
.
. ... . .
.
... ..
..
.. . ..
.
.
..
. ...
.
.. ..
.. .
.
. . ..
.. ..
.
.
.
.
..
.
.
.
.. .
.
.
.
...
.
..
.
.
.
..
.
. ...
. ..
...
.
.
..
..
... ..
.
... .
..
.
. .
.... .
.
.
.
..
.
..
..
..
.
.
. .
.
.
..
.
.
..
..
. ..
.
.
.
. .
.
.
.. .
.
.
...
..
.
.
.
...
.
... .
..
...
.
..
....
..
. ..
.
..
.
.
..
.
..
.
..
..
.
. ..
.
.
.
.
.
.
.
.. ... ..
.
.
.
.
..
..
.
.
.
.
...
..
.
.
.
. .
...
.
.
.
.
.
..
.
..
.
.. . .
.
..
.
...
.
..
....
.
..
..
.
..
...
..
.
.
..
..
...
.
.
.
.
..
. ..
.
.
.
.
.
..
.
..
.
.
..
.
.
...
.
.
.
.. .
.
.
. . .
.
. .
.
. .
.
.
.
.
.
.
.
.
.
.
...
.
.
.
..
.
.. .
.
. ..
.
...
.
.
.
.. ..
..
..
..
.
..
.
.
..
.
. .
.
..
..
.
. ..
.
.
.
.
.
.
.
.
..
.
.
.
.
.
...
.
...
.
.
...
..
...
... .
.
.
.
.
.
. ..
..
.
....
..
.
..
.
.
.
.
..
. .
.
.
...
.
.
.
.
.
..
..
.
.
..
.
.
.
...
.
.
..
...
.
.
.
.
.
.
....
.
.
.
....
.
.
.
.
....
.
. ...
.
. ..
.
.
..
.
. .
..
.
....
.
..
.
.
...
.. ..
..
..
..
...
..
.
.
..
..
. .
.
.
.. ..
....
.
... .
.
.
.
. ..
..
..
.
.
.
.
.
..
..
.
.
..
..
.
.
.
..
.
....
. ...
. .
.. .
..
...
.
.. .
.
.
. ..
..
..
..
.
..
..
. . ..
...
.
.
.. .. .
.
.
. ..
. .
.
..
...
..
.
.
..
.
.
.
.
.
. ...
.. .
.. . .
.
.
....
.
..
.
.
..
.
..
...
..
...
..
.
.
.
.
..
.
..
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
..
.
.
.. . ..
.
.
.
..
..
... .
.. ...
.
.
.
.
..
..
...
.
..
.
... ..
.
.
.
. ..
..
. .
.. ..
.
..
..
..
..
...
.
...
..
.
..
..
..
..
.. .
.
. . .
. ..
. ..
..
.. . ..
.
.
..
.
..
.
..
.
..
.
..
.
.... ... .
..
. . .
.
.
.
.
.
..
.
..
..
..
.
.
.
. .
..
.
..
.
.
.
.. ..
.
. ..
...
.
.
. ...
..
..
.
.
....
..
....
..
. . ...
.. .
...
..
.
.
.
.
.
...
..
.
..
.
.
....
.
..
..
.
.
.
.
..
. ..
..
..
..
...... .
.
.
.
.
..
.
.
.
.. .
.. .
..
.
.
...
.
..
. .
.
.
.. .
.
..
..
.. .
.. .
.
.
..... . ..
..
.
.
.
. ....
.
..
.
.
.
..
..
.
..
.
.. .. ..
..
.
..
.
..
.
.
..
..
.
.
.
.
.
.
..
..
....
.
...
.
..
.
..
.
...
.
.
. ..
..
.
.
..
.
.
.
.
.
.
..
.
...
.
.
. .. .
.
..
.
.. .
. ...... .
..
..
..
.
.
.
.
..
...
.
.
.
.
.
...
..
..
.. ..
...
.
.
..
. .
..
.
.
. ..
.
.
.
.
.
.. ..
...
.
.
.
.
.
.
.
.
.
.
.
.
.
..
....
....
.
...
..
.
. ...
.
.
. .
.
.
.
.
....
.. .
.
..
.
. .
.
.
.
. .
.
.
...
. ....
.
.
.
.
.
.
.
.
.
....
.
.
.
...
.
.
.
...
.
.
.
...
.
. ...
.
.
..
. ..
..
.
.
.
.
.
...
..
. .
..
.
. . .. . .
.
.
.
.
..
. .
.
. .
...
. .
.
.
....
.
.
...
.. . . .
. .
.
..
.
. .
..
.
.
...
.. ..
.
..
.
. ..
.
..
.
. ...
.
..
..
..
.
. .
.
..
.
.
...
.
.
.
.
.. . ..
.
.
.
.. .
..
. ..
..
.
...
.
. ..
.
.
....
..
.
..
..
.
.
..
.
.
.
.
.
. ..
...
. ..
.
.
.
.
...
.
..
.
. .
.
.
.
.
.
.
.
.
. .
.
..
.
.
.
..
.
..
. ..
.
.. ..
.
.
.
....
.
...
..
. . .
..
....
.
.
.
.
.
...
.
..
.
.
.
.
.
.
.
..
.. .
. . ....
..
.
.
.. ..
.
..
. ..
.
...
.
..
.. ..
...
.
.
.. ..
.
.
.
..
...
..
.
.
.
..
.
.
.
..
.. .
....
. ..
...
.
.
.. .
.
.
.
.
...
.
.
. .
.
.. .
. ...
.
..
...
.. ...
.
.
.
. ..
.
.
..
..
.
..
.
.
.
.
..
.....
.
...
..
. .
.....
.
...
.
.. .. .
..
.
..
..
..
.
. .
.
..
.
. ..
..
..
..
..
..
.
.
.
... ...
..
..
...
...
...
.
.
.
..
.
...
..
. . ..
..
.
..
..
..
..
.. .. .
.
.
..
..
.
..
..
.
.
...
.
...
...
. ....
... ..
.
..
.
..
.
...
.
..
.. ..
.
...
.
..
.
.
..
.
.
. .
..
....
.
.
.
..
.
...
.
.
.
..
.
.
.
.
.
.
.
.
.
..
.
.
.
.
..
..
. .
...
. . ...
...
....
.
..
.
. .
. ..
..
.. .
.
...
.
.. .
.
..
.
.
.
...
...
.
..
..
...
. ..
.
.
..
.
...
..
.
.. ... .
...
.
.
..
..
.
.
.
..
.
..
.
.
.. .
..
. ..
.
..
.
.
.
. ..
.
..
..
...
.
...
.. .
.
..
..
..
.
..
.
.
.. .
.
... .....
.
.
.
...
.
.
.
.
.... .
.
..
.
.
.
. ..
.
..
.
..
..
..
..
.
..
.
..
.
.
..
..
.
.
.
..
....
.
..
.
.
...
..
.
. ..
..
.
...
... .
..
.. .
.
. ..
...
..
.
.
.
.
.
.
.
.
.
.
...
.
. .
.
.
.
....
.
.
.
.
.
...
.
..
.....
.
.
.
.
..
..
....
.. .
.
.
.
..
.
.. .
..
..
.
.
. .
.
.
.
.
.
..
... ..
..
.
.
.
..
.
..
.
..
.
..
.. . .
..
...
.
.
.
.
.
.
.
..
.
...
. . ..
..
.
.
.
.
... . ..
..
.
.
.
... . .
. ...
..
.
.
... .
.
.
...
.
.
...
..
.
.
.
..
.
..
.
... . ..
.
.
.
.
. ....
.
..
.
.
... ..
.
..
.
...
.
.
..
.
.
..
.
..
.
.
.
..
.
..
. ... .
.. ..
.
...
.
..
..
.
.
.
...
... .
.
.
.
.
. .. ..
.
..
..
.
.
.
.
.. ..
.
.
.
.
.
.
.
..
...
.. .
.
..
.
..
.
.
..
.
.. .
.
..
..
.
..
..
..
.
.
..
.
.
.
.
. ..
.
. ..
..
...
.
.
. . ....
....
.
. .
..
.
.
...
.
. ..
..
...
.
.
..
.
.
...
.....
..
.. .
..
.
...
.
.
.
..
.
.. . .. .
.
..
.
..
.
..
... .
.
.
.
..
.
....
..
..
.
.
.
..
...
.
...
.
.
.
.
.
..
.
..
..
..
..
.
.
....
.
..
..
.
..
..
. .
.
.
.
.
.
.
..
...
.
.
...
. .
.
..
.
.
.
...
.
.
..
.
.
.
...
..
.
.
. . ..
...
.
..
. .
.
. ..
..
.
.
....
.
.... .
.
.
.
.
..
... . . .
.
...
..
.
.
. ..
.
....
.
.. .
..
.
.
...
.
..
..
..
..
.
...
.
.
.. .. ..
..
..
.
. ..
.
.
. .
...
. ..
.
. ..
.
..
...
.
.
..
..
.
.. .. .. .. .
.
.
..
...
.
..
.
..
. .
. ..
...
.
..
.
.... .
.. ..
. .
. ..
..
..
.
..... ...
.
..
.
...
. ..
. .....
..
.
.
..
.
..
..
...
...
..
.
.
..
..
...
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
..
......
.
.
....
.
.
...
.
.
..
..... ..
... .
.
.
..
..
..
.
.
. .. .. ..
. ..
.
.
.
.
.
. ..
.
...
.
.
.
..
.
.
... ..
..
.... . .
.
.
.
.
.
.
.
.
.
..
...
.
.
..
. ... .
.
. .
.
.... .. ...
..
..
. ..
.
..
.
.
..
.
.
..
.
.
.
.
.
..
.
.
.
...
..
...
.
.
.
..
.
. ..
..
..
....
..
.
. .
..
..... .
.
.
.
...
..
.
..
...... ..
.
..
.
.
..
.
... .. .
...
. .
.
... ...
...
.. .
.
...
.
.
.
. .. .
..
.
...
.
..
...
..
...
.
.
. . .
.
..
..
. ..
.
.
..
.
.. .
..
.
..
..
.
..
.
..
..
..
. ..
.
...
.
. ..
..
.
...
....
. ..
. ... .
..
..
.
.
... ..... ..
.
.
.
.
.
.
.. ..
..
.
.
..
.. ..
.... .
.. .
.
..
.. .. .
.
...
....
.
.
.
.
.. ..
...
.
.
.
.
.
.. . .
.
..
.
.
.....
.....
.
..
..
.
..
.
..
...
.
.
.
...
.
...
..
.
. .. .
..
.
.
.
.
.
.
....
..
.
.
.
.
.
.
...
.
. ..
..
.
.
.
.
. ... .
.
.
..
..
..
. ...
.. .. .
.. .
.
.
..
..
.
.. ..
. .
...
..
.. .. ... ...
..
.
...
.... . .
.
... ..
..
.
.
..
...
..
.. ... ... .
.
..
.
...
.
.
.
.
.
...... .
.
.
...
....
... .
..
.
..
..
..
.
.
.
.
.
.. ... ..
..
... ...
. .. ...
.
...
.
..
.. .
.
..
.
.. .
.
... ...
.
..... ..
....
.
.
.
..
...
.
..
.. ...
. ..
...
...
.
..
.
.
..
...
.
..
.
.
. ... ..
.
..
.
.
..
..
.
..
...
..
.
.
.
...
. .
.
.
.
...
........
. .
.
.
..
.
.
..
.
....
... .
.
. .
..
. .
.
.
..
.
. .
...
.
... .
...
...
.
. .. .
..
.
...
.. ...
.
.
.. .
... .
.. .
.
. ....
..
..
..
. .
.
.
.
.
.
..
.
.
.
.
.
.
.
.
..
...
. .
..
..
.
.
.
..
.. ..
..
...
.
..
.
.
..
..
....
..
...
.
.. ..
...
..
.
.
.
.
.. ..
.
...... .. .
.. .. ..
.
.
..
.
.. .
.
...
.
.
.... ..
.
.
. .
.
.. ... .
.. .
..
..
..
.
...
..
.
.. ..
.
.
... . ..
.
..
. .
.
.
..
.
. ...
.
......
.
..
...
.
.. ..
...
..
..
. ...
.
...
.
..
.. .
.
.
....
.
..
...
.
.
.
.
.
. ..
...
.. .
.
.. .
.
.
.
.
.. . .
.
..
.. . ......
... ... ..... ...
.
.
.. ...
...
....
.. .
. .
.
. .
.
..
...
.
.. .
..
..
.
.. .. .. ..
.
.
.
. . ..
....
.. ..
.
....
. .....
..
.
..
.
.
.
....
...
.
..
..
.
...
.
..
.....
.
.
.
.
..
.
..
..
.
....
.
.. .
.
. .
...
.
.
..
..
.
..
...
.
.
.
.
... .
.
....
.
.
.
....
..
..
..
. .. ... . .
.
. .. ... ..
.
.
... .
.
..
... ..
.. .
.
.
.
....
. .. .
. ..
.
..
.
....
.
.. . . .
.
..... . .
.
.. .
.
.
.
.
.
.. ..
..
. ..
.
..
..
..
.. ...
..
.
...
..
.
.
. ..
.
.... ..
.
.
.
.
.
.
.
..
..... ...
..
..
.....
..
.
.
..
.
. ..
..
...
..
..
.
...
..
..
..
..
..
..
.
......
. .
.
.
.
.
... .
.
.
..
.
.
. ..
.
.
.
.. .. . .
.
.
.
..
.
.
.
...
... ...
.. ...
.
.
..
.
..
.
..
.
.
. ..
..
.
. .
.
.
.. .
..
..
.
.
.
.
.
..
.
... .. .
..
...
.
..
...
..
. ...
..
...
..
.
..
..
.
.. ..... ... .
. ..
.
..
..
... . .
.
.. .
.
.
.
.
... ..
.
.
.
.
.
.
.
... .
..
. ..
.
.
.. .
... .
.. .
....
.. .
..
.... ..
.....
..
..
.
....
.
... ... . .
..
..
.
...
.
. . ..
.
. ..
.
..
.
.
..
....
..
..
..
.
....
..
. . .
...
...
.
... ... .
..
.
.
..
..
.. .
.
. .
. ..
.
.
..
......
. .
.
.
.
..
.
.
. ....
..
..
.
. .. ..
...
.
.
. ..
.....
...
... .
...
.
.
...
.
....
. .
..
.. ..
...
.
.
... . .
..
.
.. .. ..
.
..
.
.
.
.
.
. ....
.
. .. ...
... . ..
.
.
..
.
...
.
.
..
.
. .
.
.
. ..
.
.. .
.....
..
. ....
...
...
.
.
.
.
...
... . ..
..
..
.
..
.
...
.
..
.....
.. ....
...
...
.
.
..
..
..
..
.
..
..
. .
.
..
.
.... ..
.
.
.
.
. ... .
.
.
.
..
....
.
..
.
.
.
. .
..
.
...
..
.
..
.. ..
.
....
.
.
.
..
..
...
. ..
.
..
.
...
.
.. .
.
.
.
.
.
..
.
.
.
..
.
.. .
... ..
.. .
.
..
.. .
...
.....
.
.
..
...
.
.
.
. ...
..
.
.
.
.
...
. ....
... ....
...
. ..
.
.
.
.... .
..
.
.
.
..
.. ..
.
..
..
... .
.
..
..
.
.
.
.
. .... ..
.
.
.
.. .
.
..
.
.. ...
..
... . ..
.
.
.
.
.
..
.
..
..
.
. .
.
.
.
...
. . ....
..
....
.
.
..... .
... .
..
..
.
.
..
.
.
.
..
...
.. ..
.
. . ...
...
. .
.
.
..
..
.. .
.
. ... .
..
.
..
.
..
...
. .. ... ...
. ... .
.
.. .. .
.
. .
.
.. . .
.. .
. .
...
... .
.
.
....
. .
.
. .
.
.. ..
.
.
.
.
...
.
.. . .
...
.
.
.
. ..
.
..
.
.
.
.. ..
.
.
..
.
.
.
..
. ...
..
.
.
.
..
.
..
..
. .
...
..
....
. . ..
.
.
. .
.
.
.
...
... .
.
.. . ..
.
. ..
.. .. . .
.
..
.
.
.
.. .
...
.
.
....
. . ...
..
..
.
..
.
.
.
.
.
....
..
...
.....
..
..
.
..
..
..
.
..
..
. ...
.
.
.
.
.. .
.
.. . .. .
. ...
.
.
..
...
. ..
.
. ..
.
.
..
. ...
.
.
.
..
.
..
.
....
. .. .
.
.
..
. ..
....
.
. ..
....
.
. . ..
.
.
..
.. .
..
..
.
.
.
.
.
.
.
.
.
.
.
.. .. .
.
...
.
..
...
.
.
.
.
.. ..
.. .
...
.. ..
.
..
..
.. .. ...
.
.
.. .
..
...
. ..
..
. .. . .. .
.
..... . .
.....
..
....
...
..
.
.
.
.
.
...
.. .. .
..
.
.
...
..
.
..
..
..
..
.
. .....
.
. . .
.
. .
.
..
.
. .
.
..
..
.
.
.
.
.
.. ..
.
.
.. .
...
....
...
.
.
.
.
...
..
. .
.
... .... ...
.. ..
......
..
. .. ..
.. ..
....
. ..
.
. .. . .
.
..
.... .
.
.
.
.....
..
.
..
.
.
.
.
.
.
..
.
... .
...
..
.
.
.
..
.
.
.
.. .
..
.
...
..
.
... ....
.....
.
..
.
.
...
...
.
..
..
.. ..
..
.
...
.. .
...
....
.
....
..
.
.. .
.
.
.. .
. ..
.
..
.
..
.
.. . .. .. ..
.. .
..
...
.. .
...
..
...
.
...
.
..
.
..
.
...
.
.
...
.
. ....
.
.
.
.
..
..
. . ... .. . ..
.
..
..
...
...
.
. .
.
.
.
..
.
. ..
.
.
.. ...
.
... ..
.
.
.
.
. .
.
..
. ... ...
....
.
.
....
...
.
.......
.
.
..
...
...
.
..
.. ... ..
.
..
... ..
.
.
.. ..
...
.
.... .
.
....
.
..
. .. ..
.... ... .
.
...
.
.... ..
. ..
.
.. . ...
.
... .
...
.
..
.
.... ...
.
..
. ..
.
.
. ..
.
..
.
.. .
.
.
.
. ..
.
.
.
..
..
.
.
.
....
. .....
.
.
.
.. ..
...
..
.
..
.
..... ..
.
..
... . ...
.
....
...
.
...
....
..
.
.
..
.
.
..
.
.
..
... .
.
. .. .
.... . .
.
.
.. ...
..
.
.
..
..
.
...
..
.
...
... .
.
.... .
.
....
.
..
..
.
.
.
..
.
.
. .
.
. ... .. ...
..
..... .
. ...
.
. .
.. .. ...
.. ..
... ..
.
.
..
..
.
..
..
...
.
. ..
.
..
.
.
.. .. ..
..
.... ..
.. .. .. ..
..
. ..
.
. .
.
. .
..
....
.
..
.
..
..
.. . ..
.
..
...
..
..
.
.. ..
.. .
.
. ...
.
...
.. .. ...
... ..
.
..
... .
.
..
...
.
...
.
....
. ..
..
.
..... .
...
..
.. ..
. ......
. . ..
... .
.
.
... .
. .....
.
.
.. .
.
....
.
.. .
.
.
...
...
.
.
..
.
.
.. . . ... ..
.
. ..
.
...
.
..
...
.
..
..
.
.. .... ..
. ..
......
. ..
..
.
..
..
..
... ..
.
..
.... ..
.
. ..
.. .
.. .. .......
...
.
... .
.
.
.
....
. .
.
.
.......
... . ...
.
.
.
.. ...
...
. .
..
..
.
..
.
.
..
...
....
...
..
..
..
.
... .. ..
.
.
.. .. .
.
.
..
.
. ...
... ...
. .. ..
.
..
..
.
..
... .
...
.
..... ..
.
.. .
..
..
..
.
.... .
.
.
..
. .
.
. .
... ..
.
.
. .
.
.. .
.
.
. ..
.
... ..
.
..
..
.
...
..
.
.
.
...
..
.
.
.....
. ...
.
......
...
.
.
.
..
.
...
.
.
.
.. ...
.
. ... .
...
..
.
..
.. .
.
.
. ...
...
.
...
... ..
.
.. ..
...
.
.
..
.. ..
..
.
..
...
.
..
.
.
.....
.
...
..
.....
...
.
..
..
...
.
. . .. .
..
.
.
.. ..
.
.
..
.
..
.
.
.
..
...
.
.
.
..
...
... ..
...
.
...
.
.
.
..
..
.
.
..
.....
..
.
... .
.
.
.
..
.. . . .
.. .
.
.
...
.
.
.. ...
...
.
.....
...
.
..
..
.. .
...
..
.
....
.
..
.
.
..
.. .
. .
.
..
. ...
. .
..
..
.
.
.
.
. .. . .. .. . .
.
.
..
.
. ...
...
. .. ....
...
... . ..
.
..
.
..
.
.
.... .. .. ...
. .. ....
..
.
.
..
...
.. ..
.
.
.
.
. ..
..
...
..
.
..
.. ..
..
.
...
.
...
...
.
..
....
.
.
.
.
.. .
.
.
..
.
. .
.
.. .. ..
.
.
.. ..
..
.
....
.. .
.
. ..
..
.
..
.
....
..
..
.. ..
.
..
....
.
.
...
.
.
.. ...
.. ... .
.. .
...
...
..
.
.
..
..
.
..
.
.. .
.
.. . .. .
.. .
..
.
.
.
. ..
.
..
..
. .
..
..
.
. ..
..
.....
..
..
.
. .
.
.
.
..
.
...
... ... .
. ... .
....
..
.. . .
...
.
.
... .
. .
..
.
..... ..
.
.
... ..
...
.
..
..
.
.. ....
...
.
......
..
. ..... ..
.
. . ..
.
...
.
. ...
.
..
....
..
.
..
.
...
...
.
...
.. ...... .
...
.
.
.
.
.... .
. ...
..
. ....
... . .
....
.. ..
.
..
.. .
. .
..
...
.
.
..
.. ...
.
. .
... ...
..
..
.
. . ...
..
....
.
. ..
. .... ..
. .
.
....
...
.
.
. ..
...
.
.
...
....
.
..
. .. ..
.
....
..
.
. ...
.
.... .
..
.
..
.
.....
.
.. ... .
.
...
.
.
..
. ...
.. .. .... .
..
....
. ...
.
.
.
. .. .. .
.
.... .
...
...
. .
..
..
..
.
.
. ..
...
.
...
. ...
.
.
.
.
... .
.
..
...
...
...
.....
. ...
.
. .
.
... ..
...
.
..
..
. ...
...
.
...
.......
.. . .
..
.
...
..
.... ..
.
.. ...
.
. .
.
..
.
. .
..
...... ... .
.
...
.
.
.
.
... .
.. ..
.
..
.
..
.. .
..
.
.
.
..
.
.. .
...
..
.
..
.. . ....
...
.
..
.
... .
. ..
...
.
..
. . .. .
.
. ..
.
.
..
.
..
....
...
.
... . .
... ..
. . ..
.
..
..
...
.
....
.
..
..
.
.. . .
.
.. ... .
.. ...
..
.
... .
...
.
.....
..
... .
... ....
..
.
. ..
..
.
.
.
...
..
.
..
..
..
.. .... ..
........
.
. . .
.
. ...
.
... .
.
.
.. ... ...
......
...
.
..
. .. ...
..
..
. .
.
..
....
.
....
..
.
...
.
..
.
.
..
. ...
..
.
.
.
.
..
.
...
.. .. .
..
..
.. .
... .
...
.. ... ..
.
..
... ...
...
.... . .
.
.. . ..
. ...
...
..
..
. .. . .
...
..
. ...
...
. .. ....
.
...
...... .
.. ..
.
. .
..
.
.
.. .
..
..
. ..
.. .
.....
.. .
. ..
. .
..
.. . .
..
..... ..
.
..
...
.
.. .
.
.
.
... .
..
.. ...
...
..
... ..
. .
....
. .. .... .
..
.
.
.
.
.
..
.. ..
.
...
.
...
.
...
.
..
.
. .....
..
..
..
.
.. .. ..
.... .
.
...
.
..
.
.
.
.
.
..
.. . .
.
.. .
.
.
... ..
.
. ..
. .. .. .. ..
... ...
..
. ...
. .. ...
... .
. ....
..
.... ...
.
. ......
.....
..
.
.
.. ....
..
...
....
...
...
...
.
..
.
...
. ..
.
.
. . ..
....
. ..
.
.
..
.
.... . ..
..
.
.. . ...
..
.
..
.
...
..
.
.
....
...
.. ... . .
.
.
.
... .. . .
.
.. ..
.
.
..
...
.
.... . .
..
.
..
.
...
.
..
..
.
..
.
.
. ... .
.
..
.
.
.
..
.
.
. .
.. .
.
. ... .
.
.... .
.. .. .
...
.. .. ....
.. ..
.
. ... ..
...
..
...
.
.
.
.
.
. ..
.. .
.
..
..
..
.
.
...
.
.
..
..
....
....
.
...
. .
.
..
.
..
..
..
.
. .
. ...
.
.
.
.
..
. .
.
.
..
. .....
..
.
..
..
. ... ..
....
.
.
....
... .
..
.. ..
.. .
...
.
.
.
.. ..
.
......
.
..
. ..
.
...
..
..
.
..
...
... .
......
..
..
...
. ..
.. .
... ..
.
..
.. . ..
.
....
..
.
.
.
..
. . .
.
..
.
.
..
.. . .
.. .
......
....
. ...
. ....
.. ..
... .
.
. ...
.
... .. .....
..
..
....
..
........
.. ...
.. .
.
.
.
.. .
...
...
..
... .
.
..
.. .
... ...
.. .
. . ...
.. ...
. . .
..
.. . .
.
.
... ..
..
.. ... . ...
....
. ..
....
...
..
...
. ... .
.. .. .
.
.. ......
.. .
... ..
.
.... .. ..
..
.. . ... .
.....
..
..
..
.. . .
.
... .
...
..
.... ... .
. .. ..
.. ... ...
... .
.
... ..
..
.
... .
.
....
..
.
... . .
.
. ...
......
...
.
. ....
.
.
. .. ....
.
.
.
.
...
... .
.
.
....
..
..
...
.
. .....
.
...
...
..... . .
.
... .. .. .. .
..
..
. ...
.. ..
...
.
....
..
..
.. ..
.. . ...
....
....
.
.
.
...
.
...
..
.
..
..
. ..
..
..
..
..
.....
.
. .
... .
.
...
...
.
. ...
.
. .......
.
.. .
.
..
.. .
..
.... . .
. .
.
. . . ....
....
..
..
.. ..
...
...
....
..
..
.
.
... .. .
...
.
... .. . .
....
. . ..
.. ..
..
... ..
.. .. . .
..
.
... .
.
..
..
.
.
..
..
.... . ..
.. ..
.
.. .
. .. ..
..
..
....
.
.
.. .
.. ..
.....
..
...
...
.
.
..... .
.
.. .. .. .
.. .
..
.
.
.
.
. .. . .. .
. ...
..
.. .. .. .
.
..
.
..
...
. ......
... .
.
.... ........
...
...
..
.......
...
.
.
.. .
..
. .... .
. ..
. ...
..
.. . ..
.
.
. .....
...
..
...
...
..
.
.
. ..
.
..
..
... .
...
.
.
.. ...
.
.
. .
.
..
.. ..
..
..
.
..
.
.
.. ... .....
.
..
.
. ........
. .
.
..
.
..
..
. ..
.
. .. .
.
..
.. ..
.....
...
..
.
....
.
....
....
...
.. ..
..
....
..
.
.. .
.
...
. .... ...
.
... . .
.. . .. .
. .....
.
.
...
...
...
.
. .
.
. .....
... .
.
.. . ..
.
.
...
.
.. . .
. . . .....
....
..
...
....
.
..
. ..
..
..
..
. ..
. .. ..
. ..
..
..
.. ...
.
... ..
.
..
..
.. ..
.
.
. .. .
..
. .. ....
....
.
. .
.
.
..
. ....
...
.
.... .
.
. .
.. .
..
.
. . ..
..
..
.
....
..
.
... .
.
..
.
. .
.. . ....
..
.... ... ..
..
.
.. .
.
....
. ...
. ...
.....
...
.. .. .
. ...
..
...
.
...
..
. .. .
..
..
..
.. .. .
.
... .
.
.... .
..
..
.... .
.
...
.... .
.
....
.
...
....
.
.. ..
.. ..
... .
...... . ..
..
.
...
. ....
. ... ..
..
....
...
.
.... .... . .
.
....... ..
..
..
.
.
...
........ .
.
..
..
. .
..
.
.
...
..
. ....
..
..
.
.
..
.
.
.
. . ..
.
. .. . .. .. . .
.......
.. ......
.... ...
. ....
..
.
...
...
...
.
.
..
.. .. ..
. ....
.
.
...
..
. .. ..
.
.
.
.....
..
..
.
. ... .. .. ... .
..
.. ..
....
. ..
.
........ .. .
.
.
. .. .. .
..
.
.
.... .. .
... ..
..
...
.... .. ..
..
.....
.
.
.. .
. ... ...
.
...
.
..
.. .. .
.
..
..
. .. ..
......
... .. ...
. ..
..
.
. .....
.
.
....
. ..
.... .. ..
.
....
.
..
..
..
...
.
.
..
...
...
.
......
..
.
...
...
..
..
.
...
...
.
..
...
.. . .. ....
....... ...
.
.. ...
.
....
. ... ..
........
. .. .
....
....
........
..
. ..... .. ..
.
..... .
..
.
.
.... .
.. ..
.
....
........
.. ... ....
.
...
.. .....
.
.....
......
.
..
.
..... ..
...
.
.. .
.
.
...
....
.
..
.
... . . ..
..
... .. ..
..
.
...
.
..
.
..
.
..
.. ..
.
...
..
.
. ...
.
..
.
.
. .... .. .
..
.... ...
..
..
... .
. .
.. .. .....
...
. ..
.... ..
.. ...
.
....
..... ...
..
.
.
.. .
..
..
. ...
. ....
..
.
.
.
..
..
.. .
.... .
... .
.
..
...
.
. ..
..
.. .
... .
..
.. ............
... .
.
. ....
. ...
....
.. .
. .
.... .
.
. ...... ..
. ...
.
.
.. .... .
..
. .. ....
. .....
. .. ... ..
.
.
.. ....
..... .. . .
..
..
..
....
. .
....
.
.. ..
.
. .
.
.
...
..
..
.. ... .. .. ..
.
.
.
.
... ....
.....
. . ...
.
.
....
... .
...... ..
..
. ..
..
..
.. ...
........
.....
... ....
.
.....
.
.. .
.. .
.
. ...
. . .. ..
..
... ... ..
.
... .. . .
..
.
..
.
..
... .. ...
.
. .... ... .. .
.
....
. ... ......
.. ..
.
.. .
... ..
..
...
. .. ..
...
.
.... ..
.. ..
..
.
.
..
...
..
. ...
.... .
..
..
...
... .
..
. ...
..
. .... ..
. .
.
...
..
. ..
..
..... .
.
. .
.
. .. .
. ........
...
..
.
..
.. .. .
..
...
..
.
.
.
...
....
... .
. .. ....
.
.
.
. ..
.. ...
..
. .. ....
..
.
...
...
.. .. ..... ...
.
.
... ..
...
..
..
.
. ... .
..
.. ..
. . ..
...
...
.
.
...
..
...
...
.
.. .
.
..
.
.
. .
.
..
...
.
.
.
.
..
.
.....
.....
..
..
.
.
.
.
. ........ .
.
..... ..
.....
..
..... ..
..
.
. ....
.. ....
.
..
..
.
.
...
..
.. .... ...
... .
.
....
.
...
... ..
...
.. ...
.
.. .. .
..
....
... ..
.. . .... ... . ..
..
....
...
..
. ..
.. ..
.. ...
...
.. .
.
.
..
.. .. .
.
. ... ..
.. ...
..
. .. ..... .
...
. ...... .. .. .
.
... ...
..
. .... .
.
...
.. .
.
.. ..
.
.. ...
..
.....
...... ..
..
.. ..
. ... ..
...
.
.
...
.
. ....
..
.. ...
... .. .
.. ...
. ..
.. ..
.
.....
.. .
...
..
..........
.....
. .. ...
.
..
.... .
. . .
.
.
.... .. ..
..
..
.. . . ..
..
.. ...
..
.... .. .
..
. ..
.
.... .. .
.
..
....
...... ...
..... .
. .. .. ...
. ...
....
....
.
..... .
... .
.... . ....
. ...
... ..
.
... .
.
....
.. . .. ..
.... ...
...
.. .
..
..
.... ..
...
...
..
. ..
... .. ..
..
.. .
.... . ....
......
. . ...
. . .....
...
. . .. ....
.. .
...
... .
.....
...
.
.... .
..
...
.
. ..
.. ..
.. ... .
..
....
.
......
.
.. ...
.. ..
.. .
. ...
...
.
. .. ..
. ...
.
.. . .
..
.
...
. ...
.....
..
.
... .
.
..
...
..
...
. ....
.
..
.. . .. .
.
...
...
.....
..... ...
.... ..
..
...
.
....
.
........
.
. .. ...... .
...
.
..
.
. ... ..
....
.. .
...
..
...
.. ...
..
. ..
..
.. .
. .. .
.. . ..
.
.. ..... ..
..
.......
.
.... ..
.
.
..
. .. ..
.. .
.. .
. .
.
..
..
. .... ..
... ..
. .
.
...
...
.
......
.... ....
...
.
.
.....
.
. ...
..
.
..
.. .
..
.. .. .. .....
..
........ .. .
. .....
.
...
..
..
...
....
.. .
......
. ....
.
...
..
. .. ....
...
...
.. ..
.
.....
.. ..
.
..
... .
..
..
..
.
....
. ....... ..
. ..... .. .
..
... .
. .. ..
.
.
.. ..
...
...
. . ...
..... . .
.
.. .
.. .
......
.. .
...
...
..
..
.. ...
....
... ..
..
.... .
.
...
.. .. .
.
..
..
.
.
.
.. .
..
.
..
...
.
..
.. .
... ..
.. ...
..
.
.
.... .. .
..
.. .
.
.
.... ..
. ...
....
...
. ...
.
...
....
..
.
..
.
. .......
.. .. ..
..
.
.
....
.. ..
.
...
....
. ...
... .. .
.
..
....
..
... .
.....
..
....
..
.
. ...
. . ... . .....
.. ... .
... ... .. ..
.
..
.. ..
..
.. ... ..
..
....
. ..... ... .
. ..
...... .
.
..
.. ..
... ....
.....
. ..
..
.
..
.. .
.
.
. . .
.... .... .
. .... .....
. ..
..
...
...
..
. ..
.... .
.. ..
.
. . .
.....
..
.
...
..
. ...
. ...
.. ...
.. ..
.. .
..
.
. .
.
..... ...
.
.. . ..
.
.... .....
.
.. ..
... ..
....
.....
..
...........
..
..
..
.... .......
. .....
... ......
..... ..
...
.. ... .
.
...
..
.. .
.. ...
.. ..
.... .
.
...
. . .. .
.. ..
. .
....
..
..
...
. .. ..
.. .......
...
. ... .
..
... ...
.
.
..
..
... .. .
..
..
... ..
.
..
.
.. .....
...
..
.......
.
.
. .. .. ..
.
.
.
....
...
.
.
..
..
.
.. ..
..
. ..
....
.. ... ..
.... ...
....
.... .
.
.. ... . ..
.. ....
.. .. ..
...
.. .
... .. .
..
.
..
...
....
.
...... .
.
..
.....
.. ....
.. ...
. ... . .... ...
.
... .
.. ... . .. .....
.
.
.
.
....
...
...
..
....
.
. .. ..
..
.......
.
.. .... ... .
. .. ..
.
.
..
...
. ... .. ..
.
.
.
...
.
...
.. .. ..
..
..
. ... ...
...
.
..
.
.... . .. .
..
. .... .
..
.
....
.. .
... . .
.
......
. ..
.. .
... ..... ..
..
.
. ....
.. .
... .
...
...
... .
....
...
...
...
..
..... .. ........
..
. .. ......
.. ....... .
..
... ...
..
.
.. ....
.....
......
.
.. ...
. ... ..
.. ... .. ...
.
.
...... ... .
...
..
......
......
.....
.. ...
. ..
.
.
....
.
. .. .
....
.
..... . ....
.
. .... ...
.
.. .
.
......
.
.
...
.. .. .. ..
..
. .. .
.
..
..... . ..
....
.
..
..
.....
.
...
... ..
.. .... ... ..
...
..
.. .....
.. .... ..
... .... .
... ..
.. . ..
.....
.
.. ..
.
..... .. ..
... ........
..... .
...
.
..
.
...
... .. ... ..
.
...
..
.... ...
..
.... .
. ..
... ..
. ..... ...
. .....
.
..
.
...
.
...
.. .. .
..
...
... .
.
.
.
. .
.
. . ...
..
...
.. ..
..
. ..... .
..
.
. .... .
..
........
.. ..
. .. ..
..
.
.. .. ..
.
..
..
.. .. ... ..
....
.
. ..
.....
. . ..... ...
....
.
..
.. ... ...
.
..
... .... ..
..
.
....
. ..
. ...
...
. ...
..
.. . . .
.
..
..
.
....
...
...
. ....
...
..
........
..... .
...
... .
... ...
....
.
.... .... ..... ..
.
....
... ..
..
...
..... ..
. ..
......
.....
. ..
..
.
..
..
...
.
....
....
...
.
.
.
..
...
.
.. .. .
..
.. ....
..... .. .
. .. .. .. . .
.....
..
.......
.
........ .
....
. ..
. .
.
..
.. . .
.
. ..
.. ...
.. ..
.
..... . ... .. ..
. ...
.
...
.....
..
..
..
...
. ..... ...... .
...
.... .
...
.
...
..
.. ...
....
..
.
.. . ..
.. . .... .
.. ......
.
. ... ......
.
..
..
....
..
..
... .
... .
. ..... ..
. ..
..
...
... ...
...
.
.. ...
. ...
......
.
.
. .
.
..
..
...
.
......
.
..
..
..
....
...
.. .
. .....
.
.
. .. .
.. .. . ...
. .. .
.
.
.
. ..
..
.. ........
.
. ..
..
. .....
.
..
...... ..... ....
.. .
..
. .... . .
.
.....
... .
.
.
... .
.. .
...
... .
.
....
..
.... .....
...
.
..
.... ..
.... . ...
. . ..
.
. ......
..... .
..
...
....
....
.
..
c)
Figure 11: PIC simulation of resonance absorption for parametersθ = 9o, vos/c = 0.07, L/λ = 5 (kL = 10π), and nmaxe /nc = 1.5 : a)Density profile and electric field normal to the target, b) electric fieldparallel to the target, c) particle momenta, d) laser magnetic field.
Interaction with Solids Collisionless Absorption Resonance absorption 84 133
Vacuum heating: Brunel modelResonance absorption not possible if oscillation amplitudeexceeds the density scale length L, i.e. if vos/ω > L.Capacitor approximation: magnetic field of the wave is ignored;laser electric field EL has some component Ed normal to thetarget surface.
���������������������������������������
���������������������������������������
E
E n
v
kθ
d
d
L
e
∆x0 x
Figure 12: Capacitor model of the Brunel heating mechanism.
Interaction with Solids Collisionless Absorption Brunel model 85 133
Brunel model II
Driving electric fieldEd = 2EL sin θ. (52)
Field pulls a sheet of electrons out to a distance ∆x from itsinitial position. Surface number density of this sheet isΣ = ne∆x , so the electric field created between x = −∆x andx = 0 is
∆E = 4πeΣ.
Equating this to the driving field and solve for Σ:
Σ =2EL sin θ
4πe. (53)
Interaction with Solids Collisionless Absorption Brunel model 86 133
Brunel model III
When the charge sheet returns to its original position, it acquiresvelocity vd ' 2vos sin θ, where vos is the usual electron quivervelocity.Assuming these electrons are all ‘lost’ to the solid, then theaverage energy density absorbed per laser cycle is given by:
Pa =Σ
τ
mv2d
2
' 116π2
emω
E3d .
Interaction with Solids Collisionless Absorption Brunel model 87 133
Brunel model IV
Compare to the incoming laser power:
PL = cE2L cos θ/8π.
Substituting Eq. (52), we obtain the
fractional absorption rate for the Brunel mechanism:
ηa ≡Pa
PL=
4π
a0sin3 θ
cos θ, (54)
where a0 = vos/c.
Expect more absorption at large angles of incidence and higherlaser irradiance, Iλ2 ∝ a2
0.
Interaction with Solids Collisionless Absorption Brunel model 88 133
Brunel model: refinements
Two corrections to improve the model:
1 Account for the reduced driver field amplitude due toabsorption, replacing Eq. (52) by
Ed = [1 + (1− ηa)1/2]EL sin θ. (55)
2 Relativistic return velocities: replace kinetic energy with
Uk = (γ − 1)mc2
Interaction with Solids Collisionless Absorption Brunel model 89 133
Vacuum heating
Interaction with Solids Collisionless Absorption Brunel model 90 133
Fully relativistic modelGibbon, Andreev & Platanov (2011)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
0 10 20 30 40 50 60 70 80 90
(deg)
a0=0.027a0=0.085a0=0.27a0=0.85a0=2.7
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
10172 5 1018
2 5 10192 5 1020
2 5 1021
I 2 [Wcm-2 m2]
=0=6=45=60=85
Figure 13: Absorption fraction vs. angle and intensity
Interaction with Solids Collisionless Absorption Brunel model 91 133
Hot electron generation
Typical signature of collisionless laser heating – bi-Maxwellianelectron spectrum with characteristic temperatures Tc and Th.
0 100 200 300 400 500U (keV)
1
10
102
103
104
105
f(U
)dU
Te ~ 5 keV
Th ~ 75 keV
Interaction with Solids Hot Electron Generation 92 133
Short pulse Th scaling
Simple Brunel model:
TBh =
mv2d
2' 3.7I16λ
2µ (56)
EM PIC simulations:
TGBh ' 7
(I16λ
2)1/3, (57)
Relativistic j × B model (ponderomotive scaling):
TWh ' mc2(γ − 1)
' 511[(
1 + 0.73I18λ2µ
)1/2 − 1]
keV. (58)
Interaction with Solids Hot Electron Generation Scaling 93 133
Hot electron temperature: short pulses
1015 1016 1017 1018 1019 1020
I 2 (Wcm-2 m2)
2
5
102
5
1022
5
1032
5
Tho
t(k
eV)
LLE-93
IOQ-99 LULI-94
IOQ-96
LULI-97
INRS-99
MBI-97
MBI-95
RAL-96
IOQ-97
IOQ-00
RAL-99
CELV-96
CELV-96
MBI-00
STA-92 LLNL-99
LLNL-00
Th (B)Th (GB)Th (W)Th (FKL)Experiments
Figure 14: Hot electron temperature measurements in femtosecondlaser-solid experiments (squares) compared with various models: longpulse — Eq. (??); Brunel — Eq. (56); Gibbon & Bell — Eq. (57) andWilks — Eq. (58).
Interaction with Solids Hot Electron Generation Measurements of hot electron temperature 94 133
Ion acceleration
Direct acceleration in laser field inefficient, since
vi
c' eEL
miωc=
mi
mea0 ≤
a0
1836
Get relativistic ion energies for
a0 ∼ 2000
orIλ2L ≥ 5× 1024 Wcm−2µm2
Therefore need means of transmitting laser energy to ions overmany cycles→ exploit electrostatic field in plasma.
Interaction with Solids Ion acceleration 95 133
Mechanisms
1 Coulomb explosion: clusters; ponderomotive channelling ingas jets
2 Electrostatic sheath formed by hot electron cloud (TNSA)
3 Collisionless shock formation: hole boring
4 Light sail: radiation pressure on mass-limited target
Interaction with Solids Ion acceleration Mechanisms 96 133
Sheath modelElectrostatic plasma expansion into vacuum: ions initially at rest(ni = ni0), hot electrons described by Boltzmann distribution:
ne = ne0 exp(eφ/Th)
where ne0 = Zni0, and φ satisfies Poisson’s equation:
∂2φ
∂x2 =eε0
(ne − Zni)
Ion expansion is described by continuity and momentumequations: (
∂
∂t+ vi
∂
∂x
)ni = −ni
∂vi
∂x
mi
(∂
∂t+ vi
∂
∂x
)vi = −Ze
∂φ
∂x(59)
Interaction with Solids Ion acceleration Sheath model 97 133
Self-similar solutionIf the plasma stays quasineutral everywhere (ne ' Zni ), thenEqs. (59) have a self-similar solution in x/t :
Zni = ne0 exp(−x/cst − 1)
vi = cs + x/t (60)
eφ = −Te(x
cst+ 1)
where cs =√
ZTe/mi is the ion sound speed.Max ion velocity is
vf = 2cs log(τ +√τ2 + 1) (61)
where
τ =ωpi t√
2e; ωpi =
(Z 2e2ni0
ε0mi
)1/2
Interaction with Solids Ion acceleration Sheath model 98 133
Energy spectrumIon energy spectrum is given by:
dNdU
=ni0cst
(2UU0)1/2e−(2UU0)1/2
(62)
where U0 = ZkBTh.
Figure 15: Ion energy spectrum from Mora expansion model
Interaction with Solids Ion acceleration Sheath model 99 133
Hole boring
On the front side of an overdense plasma, the ponderomotiveforce will displace and compress electrons into the target,creating an electrostatic field acting on the ions.
Momentum balance across the collisionless, electrostatic shockthus formed implies:
ui = 2us = 2(
ILminic
)1/2
= 2
√ZA
me
mp
nc
ne(63)
where us is the velocity of the shock front. This leads to aquasi-monoenergetic component in the ion spectrum. See:Denavit, PRL (1992); Wilks, PRL (1992); Macchi, PRL (2005)Relativistic HB formula: Robinson, PPCF (2009).
Interaction with Solids Ion acceleration Hole boring 100 133
Light sail accelerationA mass-limited target, such as a nm-thick foil, allows nearlycomplete displacement of electrons, thereby maximizing the ESfield. A simple capacitor model suffices to determine thethreshold intensity for this scenario.The charge separation field of a foil with thickness d is:
∆E =eε0
ned
This is balanced by the net laser field at the surface,2EL = 2meωca0/e, leading to the matching condition:
a0 ' πne
nc
dλL
(64)
Under these conditions, find max ion energy Ui ∼ t1/3 – seeEsirkepov, PRL (2004).
Interaction with Solids Ion acceleration Light sail 101 133
Summary of Lecture 3
Short pulse interaction scenarios
Collisional Absorption
Collisionless Absorption
Hot Electron Generation
Ion acceleration
Interaction with Solids Ion acceleration Light sail 102 133
Mitg
lied
derH
elm
holtz
-Gem
eins
chaf
t
Physics of High IntensityLaser Plasma InteractionsPart IV: Numerical Simulation ofLaser-Plasma Interactions
20–25 June 2011 Paul Gibbon
Lecture 4: Numerical Simulation of Laser-PlasmaInteractions
Plasma modelsClassificationHierarchy
HydrodynamicsThermal transportSingle fluid modelNumerical scheme
Particle-in-Cell CodesVlasov equationParticle pusherField solverAlgorithm3D codes
Numerical Simulation of Laser-Plasma Interactions 104 133
Plasma classification
10 104 107 1010
Temperature (K)
102
106
1010
1014
1018
1024
1028
1032
Ele
ctro
nde
nsity
(cm
-3)
gaseous nebulaionosphere(F layer)
magneticfusion
electron gasin a metal
laser-plasmaslaser-plasmasJovian interior
stellar interiorstellar interior
solaratmosphere
solar corona
white dwarf
non-ideal p
lasmas
Figure 16: Plasma classification in the density-temperature plane.
Numerical Simulation of Laser-Plasma Interactions Plasma models Classification 105 133
Plasma model hierarchy
Hydrodynamics
Kinetic theory:Boltzmann equation
Fokker−Planck
Vlasov
Particle−in−Cell
Macroscopic:fluid equations
Molecular dynamics
Monte−Carlo
MHD
Steady−state
average
velocitymoments
Microscopic
collisionless
collisional
statistical
equilibrium
non−eqm
d/dt = 0
+ E, B
Figure 17: Physical basis of common plasma models andcorresponding numerical approaches.
Numerical Simulation of Laser-Plasma Interactions Plasma models Hierarchy 106 133
Numerical modelling of laser-plasma interactions
Two types of modelling dominate LPI:
1 Hydrodynamic modelling to follow the macroscopic, dynamicbehavior of the plasma, including external electric andmagnetic fields, and heating by laser or particle beams: ps-nstimescale. Good for prepulse modelling.
2 Kinetic modelling for situations in which ’non-Maxwellian’particle distributions fα(v) have to be determinedself-consistently – this is the method of choice forlaser-particle accelerator schemes!
Numerical Simulation of Laser-Plasma Interactions Plasma models Hierarchy 107 133
Hydrodynamic plasma simulation
Figure 18: Building blocks of a laser-plasma hydro-code.
Numerical Simulation of Laser-Plasma Interactions Hydrodynamics 108 133
Heating rate of plasma slab
If penetration depth of the heat wave lh < ls = c/ωp (skin depth),then the thermal transport can be neglected.Volume heated simultaneously: V ' lsπσ2
L .Setting ε = 3
2nekBTe and ∇.Φa ∼ Φa/ls, have
dTe
dt' Φa
nels, (65)
heating rate:
ddt
(kBTe) ' 4Φa
Wcm−2
(ne
cm−3
)−1( lscm
)−1
keV fs−1. (66)
Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Thermal transport 109 133
Thermal transportEnergy transport equation for a collisional plasma:
∂ε
∂t+∇.(q + Φa) = 0, (67)
where ε is the energy density, q is the heat flow and Φa = ηaΦL
is the absorbed laser flux.Huge temperature gradients are generated after a few fs: heat iscarried away from the surface into the colder target materialaccording to Eq. (67). For ideal plasmas, we write
ε =32
nekBTe
as before, and
q(x) = −κe∂Te
∂x, (68)
which is the usual Spitzer-Härm heat-flow.
Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Thermal transport 110 133
Spitzer-Härm heat-flow
Substituting for ε and q in Eq. (67) and restricting ourselves to1D by letting ∇ = (∂/∂x , 0, 0), gives a diffusion equation for Te:
32
nekB∂Te
∂t=
∂
∂x
(κe∂Te
∂x
)+∂ΦL
∂x. (69)
κe is known as the Spitzer thermal conductivity and is given by:
κe = 32(
2π
)1/2 ne
ν0m5/2T 5/2
e , (70)
where
ν0 =2πneZe4 log Λ
m2 .
Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Thermal transport 111 133
Nonlinear heat wave
0 40 80 120 160 200x (nm)
0
100
200
300
400
Te
(eV
) t = 400 fst = 300 fst = 200 fst = 100 fs
Figure 19: Nonlinear heat-wave advancing into a semi-infinite,solid-density plasma. The curves are obtained from the numericalsolution of the Spitzer heat flow equation for constant laser absorptionat the target surface (left boundary).
Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Thermal transport 112 133
Single-fluid approximation
Fluid model
∂ρ
∂t+∇·(ρu) = 0 (71)
∂ρu∂t
+∇·(ρuu) +∇P − f p = 0 (72)
∂εe
∂t+∇·
[u(εe + Pe)− κe∇Te −
Qei
γe − 1−Φa
]= 0 (73)
∂εi
∂t+∇·
[u(εi + Pi)− κi∇Ti +
Qei
γi − 1
]= 0,(74)
Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Single fluid model 113 133
Notes
Average fluid density ρ and velocity u defined thus:
ρ ≡ niM + nem ' niM,
u ≡ 1ρ
(niMui + nemue) ' ui .
The energy density εα is the sum of internal and kinetic fluidenergies:
εα =Pα
γα − 1+
12ρu2,
with γα defined as the number of degrees of freedom.Qei is the electron-ion equipartition rate
Qei =2mM
nekB(Te − Ti)
τei(75)
τei = ν−1ei is the electron-ion collision time.
Coupling to the laser Φa and f p.
Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Single fluid model 114 133
Example: laser prepulse heating solid Al target
Figure 20: Density (solid line) and temperature (dashed line) profilesfrom two interactions with pulse durations of a) 100 fs and b) 100ps.The initially cold Al target extends to x = 1 µm. a) Snapshots 0 fs, 300fs and 5 ps after the peak of the pulse; in b) snapshots are at -150 ps,-100 ps and 0 ps.
Numerical Simulation of Laser-Plasma Interactions Hydrodynamics Numerical scheme 115 133
Kinetic plasma simulation
Simplest kinetic description of a plasma uses single-particlevelocity distribution function f (r, v), evolving according to theVlasov equation:
∂f∂t
+ v· ∂f∂x
+ q(E +vc× B)· ∂f
∂p= 0. (76)
Distribution function f (r, v) is 6-dimensional – direct solutionof Eq. (76) generally impractical.
Even 1D spatial geometry still needs 2 or 3 velocitycomponents to couple to Maxwell’s equations, ie: 3- or4-dimensional code.
Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Vlasov equation 116 133
Particle-in-Cell simulationImportant method developed in the 1960s is the so-calledParticle-in-Cell (PIC) technique.Distribution function is represented instead by a large number ofdiscrete macro-particles, each carrying a fixed charge qi andmass mi .
x
p
x
pf(x,p)
i iq(x ,p )
Figure 21: Relationship between a) Vlasov and b) PIC approaches.
Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Vlasov equation 117 133
Particle pusherGeometry for simplified Lorentz equation (see Eq. 16):E = (0,Ey , 0),B = (0, 0,Bz):1/2-acceleration:
p−x = pn− 1
2x ; p−y = p
n− 12
y +∆t2
Eny
rotation:
γn = (1 + (p−x )2 + (p−y )2)1/2 ; t =∆t2
Bnz
γn ; s =2t
1 + t2
p′x = p−x + p−y t
p+y = p−y − p′xs (77)
p+x = p′x + p+
x t
1/2-acceleration:
pn+ 1
2x = p+
x ; pn+ 1
2y = p+
y +∆t2
Eny .
Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Particle pusher 118 133
Density & current gather
The density and current sources needed to integrate Maxwell’sequations are obtained by mapping the local particle positionsand velocities onto a grid as follows:
ρ(r) =∑
j
qjS(rj − r),
J(r) =∑
j
qjv jS(rj − r), j = 1...Ncell (78)
where S(rj − r) is a function describing the effective shape ofthe particles. Usually it is sufficient to use a linear weighting forS — the ‘Cloud-in-Cell’ scheme — although other more accuratemethods (eg: cubic spline) are also possible.
Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Field solver 119 133
Field solver: FDTD
Once ρ(r) and J(r) are defined at the grid points, we canproceed to solve Maxwell’s equations to obtain the new electricand magnetic fields. In 2D geometry, can define:
F + =12
(Ey + Bz)
F− =12
(Ey − Bz)
Then Faraday and Ampere laws reduce to (normalised units):
F +i+1(n + 1) = Fi(n)− ∆t
2Jn+1/2
y ,i+1/2 (forward)
F−i (n + 1) = Fi+1(n)− ∆t2
Jn+1/2y ,i+1/2 (backward) (79)
Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Field solver 120 133
PIC algorithm summary
The fields are then interpolated back to the particle positions sothat we can go back to the particle push step Eq. (??) andcomplete the cycle, see Fig. 22.
i
i i jgρ
g g
i
3. Mesh −> Particles 2. Maxwell equations
4. Push particles 1. Particles −> Mesh r v
E B g
E B
Figure 22: Iteration steps of the particle-in-cell algorithm.
Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes Algorithm 121 133
PIC codes for laser-plasma interaction studiesBecause of its simplicity and ease of implementation, thePIC-scheme sketched above is currently one of the mostimportant plasma simulation methods.
Name Authors Group(s)
OSIRIS Mori, Silva UCLA, IST LisbonVLPL Pukhov U. DüsseldorfREMP Esirkepov Kyoto, MoscowVPIC Lin, Bowers LANLEPOCH Arber, Bennett U. Warwick (CCPP)CALDER Lefebvre CEASPSC Ruhl, Brömmel LMU, FZ Jülich
Table 3: Parallel 3D, electromagnetic PIC codes
Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes 3D codes 122 133
Summary of Lecture 4
Plasma models
Hydrodynamics
Particle-in-Cell Codes
Numerical Simulation of Laser-Plasma Interactions Particle-in-Cell Codes 3D codes 123 133
Mitg
lied
derH
elm
holtz
-Gem
eins
chaf
t
Physics of High IntensityLaser Plasma InteractionsPart V: Tutorial on Particle-in-CellSimulation
20–25 June 2011 Paul Gibbon
Lecture 5: Tutorial on Particle-in-Cell Simulation
The PIC code BOPS
Prerequisites
Installation
Running BOPS
Project I: Laser wakefield accelerator
Project II: Ion acceleration – TNSA vs. RPAUseful formulae
Tutorial on Particle-in-Cell Simulation 125 133
The PIC code BOPS
BOPS is a one-and-three-halves (1 spatial, 3 velocitycoordinates: 1D3V) particle-in-cell code originally created byPaul Gibbon and Tony Bell in the Plasma Physics Group ofImperial College, London.
Based on standard algorithm for a 1D electromagnetic PICcode from Birdsall & Langdon.
Optionally employs a Lorentz ‘boost’ along the target surfaceto mimic 2D, periodic-in-y geometry, with big savings incompute time.
Applications: absorption, electron heating, high harmonicgeneration, ion acceleration
Tutorial on Particle-in-Cell Simulation The PIC code BOPS 126 133
Prerequisites
Current version: BOPS 3.4
Download site:https://trac.version.fz-juelich.de/bops
Compiler: sequential code written in Fortran 90 (Intel ifort orGNU’s gfortran)
Run scripts provided are designed for generic Unix systems:Ubuntu, SuSE, RedHat and also MacOS.
Linux and MAC users:
Unpack the tar file (name may differ) with:
tar xvfz bops3.4.tar.gz
and cd to the installation directory bops.
Tutorial on Particle-in-Cell Simulation Prerequisites 127 133
Windows users1 First install CYGWIN (www.cygwin.com). This creates a
fully-fledged Unix environment emulator under Windows. Inaddition to the default tools/packages offered during theinstallation you will also need:
Gnu compilers gcc, g77, g95 etc. (devel package)make (devel package)vi, emacs (editors package)X11 libraries if you want to generate graphics directly undercygwin (eg using gnuplot)
If you forget anything first time, just click on the Cygwininstaller icon to locate/update extra packages.
2 Download and unpack the bops3.4.tar.gz file with an archivingtool (eg PowerArchiver: www.powerarchiver.com). Put this inyour ’home’ directory under CYGWIN, e.g.:
C:\cygwin\home\<username>\
3 Open a Cygwin terminal/shell and ‘cd’ to the bops directory.
Tutorial on Particle-in-Cell Simulation Prerequisites 128 133
Installation
The directory structure resulting from unpacking the tar filesshould look like this:src/ ... containing the fortran90 source codedoc/ ... some documentation in html and pstutorial/ ... scripts and files for current tutorialbenchmarks/ ... additional sample scripts for running the codetools/ ... postprocessing tools
To compile:
Go to the source directory src, and edit the Makefile: adjustand tune the flags to match your machine type (FC=ifort orgfortran etc). Then do:
make
Tutorial on Particle-in-Cell Simulation Installation 129 133
Running BOPSOnce the code has compiled, go back up to the base (or top)directory and enter the tutorial directory. Here you will find therun scripts (suffix .sh) which launch the example simulations.These can be edited directly or copied as needed.
To run from this directory, just type eg:
./wake.sh
Or if you prefer to stay in the top directory – adjust paths inwake.sh first:
tutorial/wake.sh
Further examples can be found in benchmarks
Tutorial on Particle-in-Cell Simulation Running BOPS 130 133
Project I: Laser wakefield accelerator
1 Edit and run the script: ./wake.sh. Note that the inputparameters are normalised to laser wavelength andwavenumber.
2 Look at the printed output to check the actual laser andplasma parameters (are they what you intended?)
Plot filesThese are in run directory (foil_tnsa1) in ASCII formatand have a suffix NN.xy, where NN is the snapshotnumber. See bops.oddata for complete list
ezsi EM field Ez (S-polarized wave)nenc Electron densityexsi Electrostatic fieldpxxe Electron momenta (x − px phase space)
3 Examine the field and particle phase space plots atsuccessive time intervals (e.g.: t=100, t=200). Thegnuplot script wake.gp should help you get started.
4 How can the plasma wave amplitude | Ex |max beoptimised? Hint: try changing the electron density and/orpulse length.
5 Increase the laser amplitude to (a0 = 3) and observe thedifference in the electron phase space and plasma wave.How does the matching condition need to be altered in thisregime? Why is the amplitude of the latter reduced towardsthe back of the wake?
6 (advanced) Set up a plasma equivalent to twice thedephasing length (Eq. 37) and determine the maximumenergy of electrons trapped in the plasma wave. Can youbeat the scaling predicted by Eq. 38?
Project II: Ion acceleration – TNSA vs. RPA
1 TNSA regime.a) The script foil.sh sets up a 2 µm ‘foil’ out of frozenhydrogen (Z = A = 1), with density ne/nc = 36. This isirradiated by a linearly polarized 50 fs pulse with intensity1019 Wcm−2 .Run the script and inspect the following plots at thesnapshot times (0, 50, 100, 150) fs:
Plot files
nenc, ninc Electron & Ion densitiesexdc Cycle-averaged electrostatic fieldpxxe, pxxi Electron & ion momentafuep, fuip Electron, ion energy spectrauinc, ubac Incoming and outgoing wave energy
b) Try varying, eg: pulse amplitude or duration andcompare the scaling of the maximum ion energy againstthe theoretical prediction of Eq. (15).
2 HB regime.a) Copy the script to a new file (eg: hb.sh), switch thepolarization from linear to circular (set cpolzn='C'),change the run directory – eg: RUN=hb1 and rerun. (Whatphase space variables can you check the laser polarizationwith?) Compare the results to the TNSA case.
b) To compare with HB theory, it is convient to specify a‘square’ pulse profile. This can be done with, eg: ilas=1,trise=5, tpulse=150. Another important constraint inthis case is to avoid numerical heating arising from anunderresolved (cold) electron Debye length. Adjust the gridsize/resolution to ensure that ∆x < 0.5λD . Rerun andcompare the ion shock velocity with Eq. (63).
3 Light sail regime.a) Use the matching condition Eq. (64) to determine the foilwidth (using the same laser and target material) for whichthe light sail regime is reached. Set ilas=4 withtpulse=50, tfall=5, and ensure that the new foil isresolved by the grid. Create a new script with theseparameters and rerun. NB: you may have to increase theparameter uimax for this case!
b) Investigate the scaling behaviour of the monoenergeticion feature with intensity, pulse length etc.
c) Now repeat with a Gaussian or sin2 pulse shape.Compare the hot electron spectrum and phase space withthe flat-top case at early times: what causes thisdifference? What could you change to improve the far-fieldion sail stability for realistic pulse forms?
d) Now try the same thing with a carbon foil. Here, you willneed higher (solid) number densities and multiply chargedions (see Eq. 41). What happens to the max. ion energy?How does the energy/nucleon compare with the hydrogencase?
Formulary
Plasmafrequency ωpe = 5.64× 104n12e s−1
Critical density nc = 1021λ−2L cm−3
Debye length λD = 743 T12
e n− 12
e cm
Skin depth δ = c/ωp = 5.31× 105n− 12
e cm
Elektron-ion collision frequency νei = 2.9× 10−6neT− 32
e ln Λ s−1
Ion-ion collision frequency νii = 4.8× 10−8Z 4(
mp
mi
) 12
niT− 3
2i ln Λ s−1
Quiver momentum a0 ≡posc
mec=
(Iλ2
L
1.37× 1018Wcm−2µm2
) 12
Rel. focussing threshold Pc = 17(
nc
ne
)GW
Te in eV; ne, ni in cm−3, wavelength λL in µm
Tutorial on Particle-in-Cell Simulation Project II: Ion acceleration – TNSA vs. RPA Useful formulae 133 133