Mitglied der Helmholtz-Gemeinschaft
Physics of High Intensity
Laser Plasma Interactions
Varenna Summer School on
Laser-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
2 132
Mitglied der Helmholtz-Gemeinschaft
Physics of High Intensity
Laser Plasma Interactions
Part 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
4 132
Laser technology progress: chirped pulse
amplification
2
Intensity (W/cm )
23
Electron
energy
Relativistic ions
10
1020
Laser−nuclear physics
γ− ray sources
Particle acceleration
Fusion schemes
1 GeV
1 MeV
Relativistic optics: v osc~ c
Hard X−ray flash lamps
Hot dense matter
1 keV
1015
CPA
Field ionisation of hydrogen
Multiphoton physics
Laser medicine
1 eV
1010
1960
1 meV
1970
1980
1990
Year
2000
2010
Figure 1: Progress in peak intensity since the invention of the laser.
Definitions and thresholds
Introduction
5 132
Extreme conditions: nonlinear, strong-field science
Ordinary matter — solid, liquid or gas — rapidly ionized when
subjected to high intensity irradiation.
Electrons released are then immediately caught in the laser
field
Oscillate with a characteristic energy which dictates the
subsequent interaction physics.
Basis for laser-based particle accelerator schemes and
short-wavelength radiation sources.
Definitions and thresholds
Introduction
6 132
Electrons in intense electromagnetic fields
prehistory
Volkov (1935): electron ‘dressed’ by field – relativistic mass
increase
Definitions and thresholds
Introduction
7 132
Electrons in intense electromagnetic fields
prehistory
Volkov (1935): electron ‘dressed’ by field – relativistic mass
increase
Schwinger (1949): cyclotron radiation
Definitions and thresholds
Introduction
7 132
Electrons in intense electromagnetic fields
prehistory
Volkov (1935): electron ‘dressed’ by field – relativistic mass
increase
Schwinger (1949): cyclotron radiation
Invention of laser (1960): theoretical works on electron
dynamics
Definitions and thresholds
Introduction
7 132
Electrons in intense electromagnetic fields
prehistory
Volkov (1935): electron ‘dressed’ by field – relativistic mass
increase
Schwinger (1949): cyclotron radiation
Invention of laser (1960): theoretical works on electron
dynamics
Figure of merit q:
eEL
,
(1)
mω c
e = electron charge, m = electron mass, c = speed of light;
EL = laser electric field strength; ω = light frequency.
q=
Definitions and thresholds
Introduction
7 132
Electrons in intense electromagnetic fields
prehistory
Volkov (1935): electron ‘dressed’ by field – relativistic mass
increase
Schwinger (1949): cyclotron radiation
Invention of laser (1960): theoretical works on electron
dynamics
Figure of merit q:
eEL
,
(1)
mω c
e = electron charge, m = electron mass, c = speed of light;
EL = laser electric field strength; ω = light frequency.
q=
Ostriker & Gunn (1969) – electron dynamics in vicinity of
pulsars: q ∼ 100
Definitions and thresholds
Introduction
7 132
Field ionization
At the Bohr radius
aB =
~2
me2
the electric field strength is:
Ea
=
= 5.3 × 10−9 cm,
e
4πε0 aB2
' 5.1 × 109 Vm−1 .
Definitions and thresholds
Field ionization
(2)
8 132
Field ionization
At the Bohr radius
aB =
~2
me2
the electric field strength is:
Ea
=
= 5.3 × 10−9 cm,
e
4πε0 aB2
' 5.1 × 109 Vm−1 .
(2)
This leads to the atomic intensity :
Ia
Definitions and thresholds
=
ε0 cEa2
2
' 3.51 × 1016 Wcm−2 .
Field ionization
(3)
8 132
Field ionization
At the Bohr radius
aB =
~2
me2
the electric field strength is:
Ea
=
= 5.3 × 10−9 cm,
e
4πε0 aB2
' 5.1 × 109 Vm−1 .
(2)
This leads to the atomic intensity :
Ia
=
ε0 cEa2
2
' 3.51 × 1016 Wcm−2 .
(3)
A laser intensity of IL > Ia will guarantee ionization for any target
material, though in fact this can occur well below this threshold
value via multiphoton effects.
Definitions and thresholds
Field ionization
8 132
Tunneling ionization
Keldysh (1965) and Perelomov (1966): introduced a
parameter γ separating the multiphoton and tunneling
regimes, given by:
r
γ = ωL
2Eion
∼
IL
where
Φpond =
s
Eion
.
Φpond
e2 EL2
4mωL2
(4)
(5)
is the ponderomotive potential of the laser field.
Definitions and thresholds
Field ionization
9 132
Tunneling ionization
Keldysh (1965) and Perelomov (1966): introduced a
parameter γ separating the multiphoton and tunneling
regimes, given by:
r
γ = ωL
2Eion
∼
IL
where
Φpond =
s
Eion
.
Φpond
e2 EL2
4mωL2
(4)
(5)
is the ponderomotive potential of the laser field.
Regimes
γ < 1 ⇒ tunneling – strong fields, long wavelengths
γ > 1 ⇒ MPI – short wavelengths
Definitions and thresholds
Field ionization
9 132
Tunnelling: barrier suppression model
V(x)
−e ε x
0
x max
x
−Eion
11111
00000
00000
11111
00000
11111
00000
11111
111
000
000
111
000
111
Figure 2: a) Schematic picture of tunneling or barrier-suppression
ionization by a strong external electric field.
Definitions and thresholds
Field ionization
10 132
Barrier suppression model II
Coulomb potential modified by a stationary, homogeneous
electric field, see Fig. 13:
V (x ) = −
Ze2
− e εx .
x
⇒ suppressed on RHS of the atom, and for x xmax is lower
than the binding energy of the electron.
Definitions and thresholds
Field ionization
11 132
Barrier suppression model II
Coulomb potential modified by a stationary, homogeneous
electric field, see Fig. 13:
V (x ) = −
Ze2
− e εx .
x
⇒ suppressed on RHS of the atom, and for x xmax is lower
than the binding energy of the electron.
If the barrier falls below Eion , the electron will escape
spontaneously
⇒ barrier suppression (BS) ionization.
Set dV (x )/dx = 0 to determine the position of the barrier:
xmax = (Ze/ε),
Definitions and thresholds
Field ionization
11 132
Barrier suppression model III
Set V (xmax ) = Eion to get the threshold field strength for BS:
2
Eion
.
(6)
4Ze3
Equate critical field to the peak electric field of the laser –
appearance intensity for ions created with charge Z :
εc =
Iapp =
Definitions and thresholds
4
c 2
cEion
εc =
,
8π
128π Z 2 e6
Field ionization
(7)
12 132
Barrier suppression model III
Set V (xmax ) = Eion to get the threshold field strength for BS:
2
Eion
.
(6)
4Ze3
Equate critical field to the peak electric field of the laser –
appearance intensity for ions created with charge Z :
εc =
4
c 2
cEion
εc =
,
8π
128π Z 2 e6
Iapp =
(7)
Appearance intensity
9
Iapp ' 4 × 10
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
12 132
Appearance intensities of selected ions
Ion
Eion
(eV)
Iapp
( Wcm−2 )
H+
He+
He2+
C+
C4+
Ne+
Ne7+
Ar8+
Xe+
Xe8+
13.61
24.59
54.42
11.2
64.5
21.6
207.3
143.5
12.13
105.9
1.4 × 1014
1.4 × 1015
8.8 × 1015
6.4 × 1013
4.3 × 1015
8.6 × 1014
1.5 × 1017
2.6 × 1016
8.6 × 1013
7.8 × 1015
Table 1: BS ionization model –
Eq. (8).
Definitions and thresholds
Field ionization
13 132
Appearance intensities of selected ions
Ion
Eion
(eV)
Iapp
( Wcm−2 )
H+
He+
He2+
C+
C4+
Ne+
Ne7+
Ar8+
Xe+
Xe8+
13.61
24.59
54.42
11.2
64.5
21.6
207.3
143.5
12.13
105.9
1.4 × 1014
1.4 × 1015
8.8 × 1015
6.4 × 1013
4.3 × 1015
8.6 × 1014
1.5 × 1017
2.6 × 1016
8.6 × 1013
7.8 × 1015
Table 1: BS ionization model –
Eq. (8).
Definitions and thresholds
Field ionization
Figure 3: Auguste et al., J. Phys. B
(1992)
13 132
Relativistic field strengths
Classical equation of motion for an electron exposed to a linearly
polarized laser field E = ŷ E0 sin ω t:
dv
dt
Definitions and thresholds
'
−eE0
me
Relativistic threshold
sin ω t
14 132
Relativistic field strengths
Classical equation of motion for an electron exposed to a linearly
polarized laser field E = ŷ E0 sin ω t:
dv
dt
→v
Definitions and thresholds
=
'
−eE0
me
sin ω t
eE0
cos ω t = vos cos ω t
me ω
Relativistic threshold
(9)
14 132
Relativistic field strengths
Classical equation of motion for an electron exposed to a linearly
polarized laser field E = ŷ E0 sin ω t:
dv
dt
→v
=
'
−eE0
me
sin ω t
eE0
cos ω t = vos cos ω t
me ω
(9)
Dimensionless oscillation amplitude, or ’quiver’ velocity:
a0 ≡
Definitions and thresholds
vos
pos
eE0
≡
≡
c
me c
me ω c
Relativistic threshold
(10)
14 132
Relativistic intensity
The laser intensity IL and wavelength λL are related to E0 and ω
by:
2π c
1
IL = ε0 cE02 ; λL =
2
ω
Definitions and thresholds
Relativistic threshold
15 132
Relativistic intensity
The laser intensity IL and wavelength λL are related to E0 and ω
by:
2π c
1
IL = ε0 cE02 ; λL =
2
ω
Substituting these into (10) we find (Exercise):
a0 ' 0.85(I18 λ2µ )1/2 ,
where
I18 =
Definitions and thresholds
1018
(11)
λL
IL
; λµ =
.
−
2
µ
m
Wcm
Relativistic threshold
15 132
Relativistic intensity
The laser intensity IL and wavelength λL are related to E0 and ω
by:
2π c
1
IL = ε0 cE02 ; λL =
2
ω
Substituting these into (10) we find (Exercise):
a0 ' 0.85(I18 λ2µ )1/2 ,
where
I18 =
1018
(11)
λL
IL
; λµ =
.
−
2
µ
m
Wcm
Implies that for IL ≥ 1018 Wcm−2 , λL ' 1 µm, we will have
relativistic electron velocities, or a0 ∼ 1.
Definitions and thresholds
Relativistic threshold
15 132
Plasma definitions: the Debye length
A plasma created by field-ionization of a gas or solid will initially
be quasi-neutral. This means that the number densities of
electrons and ions with charge state Z are locally balanced:
ne ' Zni .
Definitions and thresholds
Plasma Debye length
16 132
Plasma definitions: the Debye length
A plasma created by field-ionization of a gas or solid will initially
be quasi-neutral. This means that the number densities of
electrons and ions with charge state Z are locally balanced:
ne ' Zni .
Any local disturbance in the charge distribution will be rapidly
neutralised by the lighter, faster electrons – Debye shielding:
−r /λ
shields potential around exposed charge: φD = e r D .
Definitions and thresholds
Plasma Debye length
16 132
Plasma definitions: the Debye length
A plasma created by field-ionization of a gas or solid will initially
be quasi-neutral. This means that the number densities of
electrons and ions with charge state Z are locally balanced:
ne ' Zni .
Any local disturbance in the charge distribution will be rapidly
neutralised by the lighter, faster electrons – Debye shielding:
−r /λ
shields potential around exposed charge: φD = e r D .
Debye length
λD =
Definitions and thresholds
ε0 kB Te
e2 ne
1/2
= 743
Te
eV
1/2 Plasma Debye length
ne
cm−3
−1/2
cm
(12)
16 132
Plasma classification
An ideal plasma has many particles per Debye sphere:
4π
ND ≡ ne λ3D 1.
3
Definitions and thresholds
Plasma Debye length
(13)
17 132
Plasma classification
-3
Electron density (cm )
An ideal plasma has many particles per Debye sphere:
4π
ND ≡ ne λ3D 1.
3
Definitions and thresholds
10
32
10
28
10
24
10
18
10
14
10
10
10
6
10
2
(13)
white dwarf
s
l
ea
a
sm
pla
stellar interior
id
n-
no
Jovian interior
electron gas
in a metal
laser-plasmas
solar
atmosphere
magnetic
fusion
solar corona
ionosphere
(F layer) gaseous nebula
10
4
10
Temperature (K)
10
7
10
Plasma Debye length
Figure 4: Ideal and non-ideal plasmas
10
17 132
Plasma frequency
For a thermal plasma with temperature Te one can also define a
characteristic reponse time (eg: to disturbances from external
laser fields or particle beams):
tD '
Definitions and thresholds
λD
vt
=
ε0 kB Te
e2 ne
m
·
kB Te
Plasma frequency
1/2
=
e2 ne
ε0 m e
−1/2
.
18 132
Plasma frequency
For a thermal plasma with temperature Te one can also define a
characteristic reponse time (eg: to disturbances from external
laser fields or particle beams):
tD '
λD
vt
=
ε0 kB Te
e2 ne
m
·
kB Te
1/2
=
e2 ne
ε0 m e
−1/2
.
Electron plasma frequency
ωp ≡
Definitions and thresholds
e2 ne
ε0 m e
1/2
4
' 5.6 × 10
Plasma frequency
ne
cm−3
1/2
s−1 .
(14)
18 132
Underdense and overdense plasmas
If the plasma response time is shorter than the period of a
external electromagnetic field (such as a laser), then this
radiation will be shielded out.
Definitions and thresholds
Plasma frequency
19 132
Underdense and overdense plasmas
If the plasma response time is shorter than the period of a
external electromagnetic field (such as a laser), then this
radiation will be shielded out.
Figure 5: Underdense, ω > ωp :
plasma acts as nonlinear
refractive medium
Definitions and thresholds
Plasma frequency
19 132
Underdense and overdense plasmas
If the plasma response time is shorter than the period of a
external electromagnetic field (such as a laser), then this
radiation will be shielded out.
Figure 5: Underdense, ω > ωp :
plasma acts as nonlinear
refractive medium
Definitions and thresholds
Plasma frequency
Figure 6: Overdense, ω < ωp :
plasma acts like mirror
19 132
The critical density
To make this more quantitative, consider ratio:
ωp2
e2 ne
λ2
=
·
.
ω2
ε0 me 4π 2 c 2
Definitions and thresholds
Plasma frequency
20 132
The critical density
To make this more quantitative, consider ratio:
ωp2
e2 ne
λ2
=
·
.
ω2
ε0 me 4π 2 c 2
Setting this to unity defines the wavelength for which ne = nc , or
the
Critical density
2
−3
nc ' 1021 λ−
µ cm
(15)
above which radiation with wavelengths λ > λµ will be reflected.
cf: radio waves in ionosphere.
Definitions and thresholds
Plasma frequency
20 132
Laser-generated plasmas
Target material
Capilliary discharge
Gas jet
Foam/aerogel
Frozen H
CH foil
Electron density ne
( cm−3 )
1016
1018
1021
1022
5 × 1023
ne /nc ( 800nm)
10−5
10−3
0.1 − 5
36
600
Table 2: Electron densities in typical laser-produced plasmas
Definitions and thresholds
Plasma frequency
21 132
Further reading
IC Press, London (2005)
Definitions and thresholds
Further reading
22 132
Summary of Lecture 1
Introduction
Field ionization
Relativistic threshold
Plasma Debye length
Plasma frequency
Further reading
Definitions and thresholds
Further reading
23 132
Mitglied der Helmholtz-Gemeinschaft
Physics of High Intensity
Laser Plasma Interactions
Part II: Interaction with Underdense
Plasmas
20–25 June 2011
Paul Gibbon
Lecture 2: Interaction with Underdense Plasmas
Plasma response
Cold fluid equations
EM waves
Plasma waves
EM wave propagation
Nonlinear refraction
Self focussing
SF power threshold
Ponderomotive channel formation
Plasma wave propagation
Dispersion
Numerical solutions
Wave breaking
Wakefield excitation
Electron acceleration
Bench-Top Particle Accelerators
Interaction with Underdense Plasmas
25 132
Ionized gases: when is plasma response important?
Simultaneous field ionization of many atoms produces a plasma
with electron density ne , temperature Te ∼ 1 − 10 eV. Collective
effects important if
ωp τL > 1
Interaction with Underdense Plasmas
Plasma response
26 132
Ionized gases: when is plasma response important?
Simultaneous field ionization of many atoms produces a plasma
with electron density ne , temperature Te ∼ 1 − 10 eV. Collective
effects important if
ωp τL > 1
Example (Gas jet)
τL = 100 fs, ne = 1017 cm−3 → ωp τL = 1.8
Typical gas jets: P ∼ 1bar; ne = 1018 − 1019 cm−3
Recall that from Eq.15, critical density for glass laser
nc (1µ) = 1021 cm−3 . Gas-jet plasmas are therefore
underdense, since ω 2 /ωp2 = ne /nc 1.
Interaction with Underdense Plasmas
Plasma response
26 132
Ionized gases: when is plasma response important?
Simultaneous field ionization of many atoms produces a plasma
with electron density ne , temperature Te ∼ 1 − 10 eV. Collective
effects important if
ωp τL > 1
Example (Gas jet)
τL = 100 fs, ne = 1017 cm−3 → ωp τL = 1.8
Typical gas jets: P ∼ 1bar; ne = 1018 − 1019 cm−3
Recall that from Eq.15, critical density for glass laser
nc (1µ) = 1021 cm−3 . Gas-jet plasmas are therefore
underdense, since ω 2 /ωp2 = ne /nc 1.
Exploit plasma effects for: short-wavelength radiation; nonlinear
refractive properties; high electric/magnetic fields.
Interaction with Underdense Plasmas
Plasma response
26 132
Wave propagation
The starting point for most analyses of nonlinear wave
propagation phenomena is the Lorentz equation of motion for
the electrons in a cold (Te = 0), unmagnetized plasma, together
with Maxwell’s equations.
Interaction with Underdense Plasmas
Cold fluid equations
27 132
Wave propagation
The starting point for most analyses of nonlinear wave
propagation phenomena is the Lorentz equation of motion for
the electrons in a cold (Te = 0), unmagnetized plasma, together
with 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), homogeneous
background with n0 = Zni .
Interaction with Underdense Plasmas
Cold fluid equations
27 132
Wave propagation
The starting point for most analyses of nonlinear wave
propagation phenomena is the Lorentz equation of motion for
the electrons in a cold (Te = 0), unmagnetized plasma, together
with 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), homogeneous
background with n0 = Zni .
2
Thermal motion is neglected – justified for underdense
plasmas because the temperature remains small compared
to the typical oscillation energy in the laser field.
Interaction with Underdense Plasmas
Cold fluid equations
27 132
Lorentz-Maxwell equations
Starting equations (SI units) are as follows
∂p
+ (v · ∇)p = −e(E + v × B ),
∂t
Interaction with Underdense Plasmas
Cold fluid equations
(16)
28 132
Lorentz-Maxwell equations
Starting equations (SI units) are as follows
∂p
+ (v · ∇)p = −e(E + v × B ),
∂t
∇·E =
Interaction with Underdense Plasmas
e
ε0
(n0 − ne ),
Cold fluid equations
(16)
(17)
28 132
Lorentz-Maxwell equations
Starting equations (SI units) are as follows
∂p
+ (v · ∇)p = −e(E + v × B ),
∂t
∇·E =
e
(n0 − ne ),
ε0
∂B
∇×E = −
,
∂t
Interaction with Underdense Plasmas
Cold fluid equations
(16)
(17)
(18)
28 132
Lorentz-Maxwell equations
Starting equations (SI units) are as follows
∂p
+ (v · ∇)p = −e(E + v × B ),
∂t
∇·E =
e
(n0 − ne ),
ε0
∂B
∇×E = −
,
∂t
e
∂E
c 2 ∇×B = − ne v +
,
ε0
∂t
Interaction with Underdense Plasmas
Cold fluid equations
(16)
(17)
(18)
(19)
28 132
Lorentz-Maxwell equations
Starting equations (SI units) are as follows
∂p
+ (v · ∇)p = −e(E + v × B ),
∂t
∇·E =
e
(n0 − ne ),
ε0
∂B
∇×E = −
,
∂t
e
∂E
c 2 ∇×B = − ne v +
,
ε0
∂t
∇·B = 0,
(16)
(17)
(18)
(19)
(20)
where p = γ me v and γ = (1 + p2 /me2 c 2 )1/2 .
Interaction with Underdense Plasmas
Cold fluid equations
28 132
Electromagnetic waves
To simplify matters we first assume a plane-wave geometry like
that above. A laser pulse can thus be described by the
electromagnetic fields E L = (0, Ey , 0); B L = (0, 0, Bz ).
Interaction with Underdense Plasmas
Cold fluid equations
EM waves
29 132
Electromagnetic waves
To simplify matters we first assume a plane-wave geometry like
that above. A laser pulse can thus be described by the
electromagnetic fields E L = (0, Ey , 0); B L = (0, 0, Bz ).
Exercise: From Eq. (16) one can show that the transverse
electron momentum is then simply given by:
py = eAy ,
(21)
where Ey = ∂ Ay /∂ t. This relation expresses conservation of
canonical momentum.
Interaction with Underdense Plasmas
Cold fluid equations
EM waves
29 132
The EM wave equation I
Substitute E = −∇φ − ∂ A/∂ t ; B = ∇ × A into Ampère Eq.(19):
c 2 ∇ × (∇ × A) +
∂2A
J
∂φ
=
−∇ ,
2
∂t
ε0
∂t
where the current J = −ene v .
Interaction with Underdense Plasmas
Cold fluid equations
EM waves
30 132
The EM wave equation I
Substitute E = −∇φ − ∂ A/∂ t ; B = ∇ × A into Ampère Eq.(19):
c 2 ∇ × (∇ × A) +
∂2A
J
∂φ
=
−∇ ,
2
∂t
ε0
∂t
where the current J = −ene v .
Now we use a bit of vectorial magic, splitting the current into
rotational (solenoidal) and irrotational (longitudinal) parts:
J = J ⊥ + J || = ∇ × Π + ∇Ψ
from which we can deduce (see Jackson!):
J || −
Interaction with Underdense Plasmas
1 ∂φ
∇
= 0.
c2 ∂t
Cold fluid equations
EM waves
30 132
The EM wave equation II
Now apply Coulomb gauge ∇ · A = 0 and vy = eAy /γ from (21),
to finally get:
EM wave
∂ 2 Ay
e2 ne
2 2
−
c
∇
A
=
µ
J
=
−
Ay .
y
0 y
∂t 2
ε0 m e γ
Interaction with Underdense Plasmas
Cold fluid equations
EM waves
(22)
31 132
The EM wave equation II
Now apply Coulomb gauge ∇ · A = 0 and vy = eAy /γ from (21),
to finally get:
EM wave
∂ 2 Ay
e2 ne
2 2
−
c
∇
A
=
µ
J
=
−
Ay .
y
0 y
∂t 2
ε0 m e γ
(22)
The nonlinear source term on the RHS contains two important
bits of physics:
ne = n0 + δ n → Coupling to plasma waves
γ=
q
1 + p 2 /me2 c 2
Interaction with Underdense Plasmas
→ Relativistic effects
Cold fluid equations
EM waves
31 132
Electrostatic (Langmuir) waves I
Taking the longitudinal (x)-component of the momentum
equation (16) gives:
d
e2 ∂ A2y
(γ me vx ) = −eEx −
dt
2me γ ∂ x
Interaction with Underdense Plasmas
Cold fluid equations
Plasma waves
32 132
Electrostatic (Langmuir) waves I
Taking the longitudinal (x)-component of the momentum
equation (16) gives:
d
e2 ∂ A2y
(γ me vx ) = −eEx −
dt
2me γ ∂ x
We can eliminate vx using Ampère’s law (19)x :
0=−
Interaction with Underdense Plasmas
e
ε0
ne vx +
Cold fluid equations
∂ Ex
,
∂t
Plasma waves
32 132
Electrostatic (Langmuir) waves I
Taking the longitudinal (x)-component of the momentum
equation (16) gives:
d
e2 ∂ A2y
(γ me vx ) = −eEx −
dt
2me γ ∂ x
We can eliminate vx using Ampère’s law (19)x :
0=−
e
ε0
ne vx +
∂ Ex
,
∂t
while the electron density can be determined via Poisson’s
equation (17):
ε0 ∂ E x
.
ne = n0 −
e ∂x
Interaction with Underdense Plasmas
Cold fluid equations
Plasma waves
32 132
Electrostatic (Langmuir) waves II
The above (closed) set of equations can in principle be solved
numerically. For the moment, we simplify things by linearizing
the plasma quantities:
ne
' n0 + n1 + ...
vx
' v1 + v2 + ...
and neglect products like n1 v1 etc. This finally leads to:
Driven plasma wave
ωp2
∂2
+
∂t 2
γ0
Interaction with Underdense Plasmas
!
Ex = −
ωp2 e
2me γ02
Cold fluid equations
∂ 2
A
∂x y
Plasma waves
(23)
33 132
Electrostatic (Langmuir) waves II
The above (closed) set of equations can in principle be solved
numerically. For the moment, we simplify things by linearizing
the plasma quantities:
ne
' n0 + n1 + ...
vx
' v1 + v2 + ...
and neglect products like n1 v1 etc. This finally leads to:
Driven plasma wave
ωp2
∂2
+
∂t 2
γ0
!
Ex = −
ωp2 e
2me γ02
∂ 2
A
∂x y
(23)
The driving term on the RHS is the relativistic ponderomotive
force, with γ0 = (1 + a02 /2)1/2 .
Interaction with Underdense Plasmas
Cold fluid equations
Plasma waves
33 132
Cold plasma wave equations: recap
Electromagnetic wave
∂ 2 Ay
e2 ne
2 2
−
c
∇
A
=
µ
J
=
−
Ay
y
0
y
∂t 2
ε0 m e γ
Electrostatic (Langmuir) wave
ωp2
∂2
+
∂t 2
γ0
Interaction with Underdense Plasmas
!
Ex = −
ωp2 e
2me γ02
Cold fluid equations
∂ 2
A
∂x y
Plasma waves
34 132
Cold plasma wave equations: recap
Electromagnetic wave
∂ 2 Ay
e2 ne
2 2
−
c
∇
A
=
µ
J
=
−
Ay
y
0
y
∂t 2
ε0 m e γ
Electrostatic (Langmuir) wave
ωp2
∂2
+
∂t 2
γ0
!
Ex = −
ωp2 e
2me γ02
∂ 2
A
∂x y
These coupled fluid equations describe a vast range of nonlinear
laser-plasma interaction phenomena: parametric instabilities,
self-focussing, channelling, wakefield excitation, harmonic
generation, ...
Interaction with Underdense Plasmas
Cold fluid equations
Plasma waves
34 132
Dispersion properties: EM waves
This time we switch the plasma oscillations off (ne = n0 ) in
Eq.(22) and look for solutions:
Ay = A0 sin(ω t − kx ),
Interaction with Underdense Plasmas
EM wave propagation
35 132
Dispersion properties: EM waves
This time we switch the plasma oscillations off (ne = n0 ) in
Eq.(22) and look for solutions:
Ay = A0 sin(ω t − kx ),
to obtain
ω2 =
Interaction with Underdense Plasmas
ωp2
+ c2k 2.
γ0
EM wave propagation
(24)
35 132
Dispersion properties: EM waves
This time we switch the plasma oscillations off (ne = n0 ) in
Eq.(22) and look for solutions:
Ay = A0 sin(ω t − kx ),
to obtain
ω2 =
ωp2
+ c2k 2.
γ0
(24)
From this relation we can derive a
Nonlinear refractive index
r
η=
Interaction with Underdense Plasmas
c2k 2
ω2
=
ωp2
1−
γ0 ω 2
EM wave propagation
!1/2
(25)
35 132
Nonlinear refraction effects
Have so far assumed plane wave approximation for laser pulse –
’photon bullet’
Real laser pulses are created with focusing optics & are subject
to:
1
diffraction due to finite focal spot σL :
ZR = 2πσL2 /λ
2
ionization effects: refraction due to radial density gradients
3
relativistic self-focusing. Power threshold:
Pc ' 17
4
ω0
ωp
2
GW,
(26)
ponderomotive channelling
All nonlinear effects important for PL > 2TW
Interaction with Underdense Plasmas
EM wave propagation
Nonlinear refraction
36 132
Relativistic self-focussing: Geometric optics
Consider laser beam with a radial profile
a(r ) = a0 exp(−r 2 /2σL2 ),
spot size σ0 just inside a region of uniform, underdense plasma,
see Fig. 7.
∆L
plasma
R α
σ
laser
α
θ
Z
Z
a)
b)
Figure 7: a) diffraction, b) self-focusing
Interaction with Underdense Plasmas
EM wave propagation
Self focussing
37 132
Geometric optics picture: power threshold
Beam spreading due to diffraction will be cancelled by
self-focusing effects if θ = α, (Exercise!),
a02
ω σ 2
p L
c
≥ 8.
(27)
This represents a power threshold, since the laser power
PL ∝ a02 σL2 . In numbers:
PL > 9
ω
ωp
2
GW
This is ∼ ×2 too low because we didn’t take beam profile into
account.
Interaction with Underdense Plasmas
EM wave propagation
SF power threshold
38 132
Focussing threshold – practical units
Litvak, 1970; Max et al.1974, Sprangle et al.1988
Relation between laser power and critical power:
PL
mω c 2 c 2 c Z ∞
0
=
2π ra2 (r )dr
e
ωp
2 0
2
ω
1 m 2 5
c 0
=
P̃
2 e
ωp
2
ω
P̃ GW.
' 0.35
ωp
The critical power P˜c = 16π thus corresponds to:
Power threshold for relativistic self-focussing
Pc ' 17.5
Interaction with Underdense Plasmas
ω
ωp
2
EM wave propagation
GW,
(28)
SF power threshold
39 132
Focussing threshold – example
Critical power
Pc ' 17.5
ω
ωp
2
GW,
(29)
Example
λL = 0.8µm, ne = 1019 cm−3
ω 2
ne
1019
p
⇒
=
=
= 6 × 10−3
nc
ω
1.6 × 1021
⇒ Pc = 2.6TW
Interaction with Underdense Plasmas
EM wave propagation
SF power threshold
40 132
Transverse plasma response
Cigar-shaped pulse: take ∇ = ∇⊥ , and apply Poisson’s
equation (φ and n normalized as before)
∇2⊥ φ = kp2 (n − 1),
to obtain density perturbation:
Electron density n(r)
n = 1 + kp−2 ∇2⊥ γ.
(30)
1.5
1.2
0.9
0.6
a0=0.2
a0=0.5
a0=1.0
a0=2.0
0.3
0.0
0
1
2
3
4
5
kpr
Interaction with Underdense Plasmas
EM wave propagation
Ponderomotive channel formation
41 132
Cavitation condition
Consider Gaussian pulse profile a(r ) = a0 exp(−r 2 /2σ 2 ). After
time-averaging over the laser period, the density depression
term in cylindrical coordinates can be written:
1 4a02
1 2 2
∇⊥ a =
∇⊥ γ =
4γ
4γ σ 2
2
Interaction with Underdense Plasmas
EM wave propagation
r2
σ2
− 1 exp(−r 2 /σ 2 ).
Ponderomotive channel formation
42 132
Cavitation condition
Consider Gaussian pulse profile a(r ) = a0 exp(−r 2 /2σ 2 ). After
time-averaging over the laser period, the density depression
term in cylindrical coordinates can be written:
1 4a02
1 2 2
∇⊥ a =
∇⊥ γ =
4γ
4γ σ 2
2
r2
σ2
− 1 exp(−r 2 /σ 2 ).
The deepest depression is on the laser axis at r = 0, giving
Cavitation condition, (n = 0):
I18 λ2µ >
1
n18 σµ2 ,
20
(31)
– quite easily fulfilled with TW lasers.
Interaction with Underdense Plasmas
EM wave propagation
Ponderomotive channel formation
42 132
Relativistic beam propagation
Numerical solution of NLSE with an initial radial Gaussian beam
profile with σ0 = 7.5 µm and pump strengths a0 .
Beam powers P /Pc : 0.5, 1.0 and 3.0 respectively.
30
vacuum
P=Pc/2
P=Pc
P=3Pc
Beam radius r ( m)
25
20
15
10
5
0
0
1
2
3
4
z (mm)
Interaction with Underdense Plasmas
EM wave propagation
Ponderomotive channel formation
43 132
Plasma (Langmuir) wave propagation
Without the laser driving term (Ay = 0), Eq.(23) describes linear
plasma oscillations with solutions
Ex = Ex0 sin(ω t ),
Interaction with Underdense Plasmas
Plasma wave propagation
Dispersion
44 132
Plasma (Langmuir) wave propagation
Without the laser driving term (Ay = 0), Eq.(23) describes linear
plasma oscillations with solutions
Ex = Ex0 sin(ω t ),
giving the dispersion relation:
−ω 2 + ωp2 = 0.
(32)
The linear eigenmode of a plasma has ω = ωp .
Interaction with Underdense Plasmas
Plasma wave propagation
Dispersion
44 132
Plasma (Langmuir) wave propagation
Without the laser driving term (Ay = 0), Eq.(23) describes linear
plasma oscillations with solutions
Ex = Ex0 sin(ω t ),
giving the dispersion relation:
−ω 2 + ωp2 = 0.
(32)
The linear eigenmode of a plasma has ω = ωp .
Including finite temperature Te > 0 yields the Bohm-Gross
relation:
ω 2 = ωp2 + 3vt2 k 2 .
Interaction with Underdense Plasmas
Plasma wave propagation
Dispersion
(33)
44 132
Numerical solutions – linear Langmuir wave
Numerical integration of the electrostatic wave equation on slide
1 for vmax /c = 0.2
1.5
u
E
ne/n0
u, E, ne/n0
1.0
0.5
0.0
-0.5
0.0
3.1
6.2
p
NB: electric field and density 90o out of phase
Interaction with Underdense Plasmas
Plasma wave propagation
Numerical solutions
45 132
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
8
8
a)
6
u, E, ne/n0
u, E, ne/n0
6
4
2
0
b)
u
E
ne/n0
4
2
0
-2
0
2
4
6
8
-2
0.0
p
3.1
6.2
p
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
46 132
Maximum field amplitude - wave-breaking limit
For relativistic phase velocities, find
Emax ∼ mωp c /e
– wave-breaking limit – Dawson (1962), Katsouleas (1988).
Interaction with Underdense Plasmas
Plasma wave propagation
Wave breaking
47 132
Maximum field amplitude - wave-breaking limit
For relativistic phase velocities, find
Emax ∼ mωp c /e
– wave-breaking limit – Dawson (1962), Katsouleas (1988).
Example
me
c
= 9.1 × 10−28 g
= 3 × 1010 cms−1
ωp = 5.6 × 104 (ne /cm−3 )1/2
e = 4.8 × 10−10 statcoulomb
Ep
∼ 4 × 10
Interaction with Underdense Plasmas
8
ne
18
10 cm−3
Plasma wave propagation
1/2
V m−1
Wave breaking
47 132
Wakefield excitation
f pond
+
t1
+
+
f pond
t2
+
+
+
eEx
f pond
+
t3
+
+
eEx
Interaction with Underdense Plasmas
Wakefield excitation
48 132
Resonance condition
The amplitude of the longitudinal oscillation will be enhanced if
the pulse length is roughly matched to the plasma period:
τL ' ωp−1 .
Interaction with Underdense Plasmas
Wakefield excitation
49 132
Resonance condition
The amplitude of the longitudinal oscillation will be enhanced if
the pulse length is roughly matched to the plasma period:
τL ' ωp−1 .
Example
What plasma density do we need to match a 100 fs pulse?
1/2
ωp ' 5 × 104 ne
Interaction with Underdense Plasmas
Wakefield excitation
s −1
49 132
Resonance condition
The amplitude of the longitudinal oscillation will be enhanced if
the pulse length is roughly matched to the plasma period:
τL ' ωp−1 .
Example
What plasma density do we need to match a 100 fs pulse?
1/2
ωp ' 5 × 104 ne
s −1
Matching condition:
−2
ne ' 4 × 1014 τps
cm−3
For 100 fs, need ne = 4 × 1016 cm−3 .
Interaction with Underdense Plasmas
Wakefield excitation
49 132
Numerical solution: small laser amplitude
wakefield
0.06
E
n
pump
0.04
0.02
0.0
-0.02
0
5
10
15
20
25
kp
Interaction with Underdense Plasmas
Wakefield excitation
50 132
Numerical solution: resonance condition (small
amplitudes)
0.005
Emax
0.004
0.003
0.002
0.001
0.0
0
5
10
kp
Interaction with Underdense Plasmas
15
20
L
Wakefield excitation
51 132
Numerical solution: large laser amplitude
3
pump
E
n
wakefield
2
1
0
-1
0
2
4
6
8
10
12
14
16
18
20
kp
Interaction with Underdense Plasmas
Wakefield excitation
52 132
Wake amplitude scaling in nonlinear regime
Murusidze & Berzhiani, 1990
Analytical solution possible for a square pump in the limit βg → 1
⇒ Scaling of the wake-variable maxima:
2
φmax ∼ γ⊥
−1
2
γ −1
Emax ∼ ⊥
γ⊥
4
γ −1
pmax ∼ (γ u )max = ⊥ 2
2γ⊥
(34)
where γ⊥ = (1 + a2 )1/2 as on p. ??.
Interaction with Underdense Plasmas
Wakefield excitation
53 132
2D wakefield excitation
Interaction with Underdense Plasmas
Wakefield excitation
54 132
Electron acceleration by wakefields
Conventional synchrotrons and LINACS operate with field
gradients limited to around 100 MVm−1 .
Plasma is already ionized; can theoretically sustain a field 104
times larger, given by:
Ep
=
me c ωp
ε
e
1/2
' n18 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
55 132
Laser-electron accelerator
Tajima & Dawson, 1979
Laser-driven wakefields must propagate with velocities
approaching the speed of light (vp = vg < c).
Plasma wave has a phase velocity:
vp = c
ωp2
1− 2
ωo
!1
2
1
'c 1− 2 ,
2γp
(36)
where γp = ω02 /ωp2 .
Interaction with Underdense Plasmas
Wakefield excitation
Electron acceleration
56 132
Acceleration length
A relativistic electron (v ' c) trapped in such a wave will be
accelerated over at most half a wavelength in the wave-frame,
after which it starts to be decelerated.
Effective acceleration length:
La =
λp c
2(c − vp )
' λp γp2
=
ω2
λp
ωp2
−3/2 −2
λ µm
' 3.2 n18
Interaction with Underdense Plasmas
Wakefield excitation
cm.
Electron acceleration
(37)
57 132
Maximum energy gain
Combine Eq. (35) and Eq. (37) to obtain the maximum energy
gain:
∆U = eEp .La
mω c ω 2 2π c
p
ε 2
= e
e
ωp ωp
2
ω
ε mc 2
= 2π
ωp
−1 −2
' 3.2 n18
λ µm GeV.
Interaction with Underdense Plasmas
Wakefield excitation
Electron acceleration
(38)
58 132
Limiting factors
In principle, TW laser is capable of accelerating an electron to
5 GeV in a distance of 5 cm through a plasma with density
1018 cm−3 .
Spoiling factors:
Diffraction: typically have ZR La , so some means of
guiding 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
59 132
Bench-Top Particle Accelerators
Standard electron acceleration in fast plasma wave (recap):
Acceleration length
−3/2 −2
λ µm
La ' 3.2 n18
cm
Energy gain
−1 −2
∆U ' 3.2 n18
λ µm GeV
Interaction with Underdense Plasmas
Wakefield excitation
Bench-Top Particle Accelerators
60 132
Acceleration mechanisms
Large variety of attributed acceleration mechanisms in
experiments (long and short pulse):
self-modulated
forced-wave
wave-breaking
guided
bubble-regime
GeV milestone reached September 2006 (LBL). More to come
on Wed–Fri ....
Interaction with Underdense Plasmas
Wakefield excitation
Bench-Top Particle Accelerators
61 132
What mechanisms are at work here?
Interaction with Underdense Plasmas
Wakefield excitation
Bench-Top Particle Accelerators
62 132
Livingstone chart for laser-plasma electron
accelerators
Max. electron energy (MeV)
2
10
LBL
3
5
RAL
LOA
2
10
RAL
2
UCLA
2
CUOS
NRL
CUOS
ILE
ILE
LBL
RAL (Astra)
RAL
5
LOA
MPQ
10
5
2
1
1980
LULI
INRS
UCLA
IC/RAL
1985
1990
1995
2000
2005
2010
Year
Figure 8: Open circles represent early beat-wave experiments; filled
circles single pulse wakefield experiments; triangles
quasi-monoenergetic electron beams
Interaction with Underdense Plasmas
Wakefield excitation
Bench-Top Particle Accelerators
63 132
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
64 132
Mitglied der Helmholtz-Gemeinschaft
Physics of High Intensity
Laser Plasma Interactions
Part III: Interaction with Solids
20–25 June 2011
Paul Gibbon
Lecture 3: Interaction with Solids
Short pulse interaction scenarios
Collisional Absorption
Normal skin effect
Collisionless Absorption
Resonance absorption
Brunel model
Hot Electron Generation
Scaling
Measurements of hot electron temperature
Ion acceleration
Mechanisms
Sheath model
Hole boring
Light sail
Interaction with Solids
66 132
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 )
Interaction with Solids
Short pulse interaction scenarios
67 132
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
67 132
Typical interaction scenario: I
Creation of critical surface
Comination of field and collisional ionization over the first few
laser cycles rapidly creates a surface plasma layer with a density
many times the critical density nc .
ω2 =
4π e2 nc
,
me
(39)
where e and me are the electron charge and mass respectively.
Interaction with Solids
Short pulse interaction scenarios
68 132
Typical interaction scenario: I
Creation of critical surface
Comination of field and collisional ionization over the first few
laser cycles rapidly creates a surface plasma layer with a density
many times the critical density nc .
ω2 =
4π e2 nc
,
me
(39)
where e and me are the electron charge and mass respectively.
In practical units (see Eq. 15):
21
nc ' 1.1 × 10
Interaction with Solids
λ
µm
−2
Short pulse interaction scenarios
cm−3 .
(40)
68 132
Interaction scenario: II
Ionization degree
Example
Al has 3 valence electrons; 6 more can be released for a few
hundred eV. The electron density is given by:
ne = Z ∗ ni =
effective ion charge:
atomic number:
Avogadro number:
mass density:
electron density:
density contrast (1 µm):
Interaction with Solids
Z ∗ NA ρ
.
A
(41)
Z∗ = 9
A = 26
NA = 6.02 × 1023
ρ = ρsolid = 1.9 g cm−3
ne = 4 × 1023 cm−3
ne /nc ' 400
Short pulse interaction scenarios
69 132
Interaction scenario: II
Ionization degree
Example
Al has 3 valence electrons; 6 more can be released for a few
hundred eV. The electron density is given by:
ne = Z ∗ ni =
effective ion charge:
atomic number:
Avogadro number:
mass density:
electron density:
density contrast (1 µm):
Interaction with Solids
Z ∗ NA ρ
.
A
(41)
Z∗ = 9
A = 26
NA = 6.02 × 1023
ρ = ρsolid = 1.9 g cm−3
ne = 4 × 1023 cm−3
ne /nc ' 400
Short pulse interaction scenarios
69 132
Interaction scenario: III
Heating
Target is heated via electron-ion collisions to 10s or 100s of eV
depending on the laser intensity.
The plasma pressure created during heating causes ion blow-off
(ablation) at the sound speed:
cs
=
Z ∗ kB Te
mi
' 3.1 × 10
7
1/2
Te
keV
1/2 Z∗
A
1/2
cm s−1 ,
(42)
where kB is the Boltzmann constant, Te the electron temperature
and mi the ion mass.
Interaction with Solids
Short pulse interaction scenarios
70 132
Interaction scenario: IV
Expansion
Because of ion ablation, density profile formed is exponential
with scale-length:
L = cs τL
' 3
Interaction with Solids
Te
keV
1/2 Z∗
A
Short pulse interaction scenarios
1/2
τfs Å.
(43)
71 132
Interaction scenario: IV
Expansion
Because of ion ablation, density profile formed is exponential
with scale-length:
L = cs τL
' 3
Te
keV
1/2 Z∗
A
1/2
τfs Å.
(43)
Example
100 fs Ti:sapphire pulse on Al foil heats the target to a few
hundred eV → plasma with scale-length L/λ= 0.01–0.1. (cf:
100-1000 for ICF plasmas).
Interaction with Solids
Short pulse interaction scenarios
71 132
Collisional absorption
Modelled using Helmholtz wave equations: standard method for
electromagnetic wave propagation in an inhomogeneous plasma
– see books by Ginzburg, Kruer.
Start from Maxwell’s equations with small field amplitudes and a
non-relativistic fluid response including collisional damping:
m
v
∂v
= −e(E + ×B ) − mνei v ,
∂t
c
(44)
where νei is the electron-ion collision frequency.
Interaction with Solids
Collisional Absorption
72 132
Collisional absorption
Modelled using Helmholtz wave equations: standard method for
electromagnetic wave propagation in an inhomogeneous plasma
– see books by Ginzburg, Kruer.
Start from Maxwell’s equations with small field amplitudes and a
non-relativistic fluid response including collisional damping:
m
v
∂v
= −e(E + ×B ) − mνei v ,
∂t
c
(44)
where νei is the electron-ion collision frequency.
Physically arises from binary collisions, resulting in a frictional
drag on the electron motion.
Interaction with Solids
Collisional Absorption
72 132
Electron-ion collisional frequency
Spitzer-Härm
Collision rate:
νei
=
4(2π)1/2 ne Ze4
ln Λ
3
3
m2 vte
−3/2
' 2.91 × 10−6 Zne Te
ln Λ s−1 .
(45)
Z = number of free electrons per atom
ne = electron density in cm−3
Te = temperature in eV
ln Λ is the Coulomb logarithm, with usual limits, bmin and bmax ,
of the electron-ion scattering cross-section.
Interaction with Solids
Collisional Absorption
73 132
Coulomb logarithm
Limits are determined by the classical distance of closest
approach and the Debye length respectively, so that:
Λ=
bmax
kB Te
9ND
= λD . 2 =
,
bmin
Ze
Z
where
λD =
kB Te
4π ne e2
1/2
=
vte
ωp
,
(46)
(47)
and
4π 3
λ ne
3 D
is the number of particles in a Debye sphere.
ND =
Interaction with Solids
Collisional Absorption
74 132
Absorption in steep density profiles: skin effect
Density profile can be approximated by a Heaviside step
function:
n0 (x ) = n0 Θ(x ),
Interaction with Solids
Collisional Absorption
Normal skin effect
75 132
Absorption in steep density profiles: skin effect
Density profile can be approximated by a Heaviside step
function:
n0 (x ) = n0 Θ(x ),
giving a dielectric constant (cf: dispersion relation Eq. 24):
ε(x ) ≡
Interaction with Solids
=1−
ωp2
Θ(x ).
ω 2 (1 + i νei /ω)
Collisional Absorption
Normal skin effect
c2k 2
ω2
(48)
75 132
Solution for laser field I
For normally incident light, the transverse electric field has the
solution
Ez =
2E0 sin(kx cos θ + φ), x < 0
E (0) exp(−x /ls ),
x ≥0
(49)
where ls ' c /ωp is the collisionless skin-depth, k = ω/c, E0 is
the amplitude of the laser field and φ a phase factor.
Interaction with Solids
Collisional Absorption
Normal skin effect
76 132
Solution for laser field I
For normally incident light, the transverse electric field has the
solution
Ez =
2E0 sin(kx cos θ + φ), x < 0
E (0) exp(−x /ls ),
x ≥0
(49)
where ls ' c /ωp is the collisionless skin-depth, k = ω/c, E0 is
the amplitude of the laser field and φ a phase factor.
Matching vacuum and solid solutions at the boundary x = 0
gives:
ω
cos θ
ωp
ω
tan φ = −ls cos θ.
E (0) = 2E0
c
Interaction with Solids
Collisional Absorption
Normal skin effect
76 132
Solution for laser field I
6
2
By
2
Ez
Field intensity
5
n0/nc
4
3
2
2
1
0
-4
2
E (0) = 2E0 nc/n0
-3
-2
-1
0
1
2
kx
Interaction with Solids
Collisional Absorption
Normal skin effect
77 132
Reflectivity: Drude model
Example
Al: Z ∗ = 3, ne ' 2 × 1023 cm−3 , λL = 0.8 µm , ne /nc ' 100.
1.0
Helmholtz
Fresnel
= 1-R
0.6
a
0.8
0.4
0.2
0.0
0
15
30
45
o
60
75
90
()
Figure 9: Angular absorption for a step-profile with ne /nc = 100 and
ν/ω = 5 calculated analytically from the Fresnel equations (solid) and
numerically from the Helmholtz wave equations (dotted).
Interaction with Solids
Collisional Absorption
Normal skin effect
78 132
Collisional frequency turn-off
quiver velocity correction
Effective collision frequency reduced by quiver motion in laser
field
v3
νe ' νei 2 te 2 3/2 .
(50)
(vos + vte )
Interaction with Solids
Collisionless Absorption
79 132
Collisional frequency turn-off
quiver velocity correction
Effective collision frequency reduced by quiver motion in laser
field
v3
νe ' νei 2 te 2 3/2 .
(50)
(vos + vte )
A temperature of 1 keV corresponds to a thermal velocity
vte ' 0.05, so collisional absorption starts to turn off for
irradiances I λ2 ≥ 1015 Wcm−2 µm2 .
Interaction with Solids
Collisionless Absorption
79 132
Collisionless absorption mechanisms
What other absorption mechanisms can couple laser energy to a
hot, solid-density target?
Interaction with Solids
Collisionless Absorption
80 132
Collisionless absorption mechanisms
What other absorption mechanisms can couple laser energy to a
hot, 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)
Interaction with Solids
Collisionless Absorption
80 132
Collisionless absorption mechanisms
What other absorption mechanisms can couple laser energy to a
hot, 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 with
energies Th ∼ keV–MeV.
Interaction with Solids
Collisionless Absorption
80 132
Collisionless resonance absorption
n e (x)
nc
n c cos2 θ
θ
y
B
E
s
B
x
k
E
p
Figure 10: Standard picture of resonance absorption: a p-polarized
light wave tunnels through to the critical surface (ne = nc ) and drives
up a plasma wave. This is damped by particle trapping and wave
breaking at high intensities.
Interaction with Solids
Collisionless Absorption
Resonance absorption
81 132
Resonance absorption: Denisov function
Self-similar dependence on the parameter ξ = (kL)1/3 sin θ,
kL 1.
1.4
1.2
( )
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.5
1.0
1.5
=(kL)
Interaction with Solids
Collisionless Absorption
1/3
2.0
2.5
3.0
sin
Resonance absorption
82 132
Absorption rate for long scalelengths
To a good approximation,
φ(ξ) ' 2.3ξ exp(−2ξ 3 /3),
(51)
and the fractional absorption is given by:
ηra =
1 2
Φ (ξ).
2
Behavior nearly independent of the damping mechanism.
Interaction with Solids
Collisionless Absorption
Resonance absorption
83 132
Kinetic simulation of resonance absorption
Particle-in-Cell
0.15
2.0
a)
1.0
0.1
0.05
0.5
0.0
0
2
4
6
b)
Ey
ne/nc
x
1.5
0.0
8 10
0
2
x/
0.4
...
...
......
...............
.......
............
........ .. . .
..
.. . . ....... ... ..................... .. ............... ...... ...............................................
..... . . .................. ................................................
. ..........
........................................
.....................................
..................
................
.......................................................
. ... ... .....................
.........................................
...........
..........
. ..............
...
.......................
.......................
.........
...........
............
.............................................. ...............
.......................................
...
................
......................
.........................
...........................................
......
....
........
.....
.......
..........
.
......
.
.........
.............
................................
.....
..............
.
.
.
.......................
.
..
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.. .............
.
.
.
...
.
.
.
.
.
.
.
.
.
.
.
.
............. .......................
....
................
.......................
.......................
...........................
........................
............
......
...................................................
.............................
......
.
.
.
.
.
.
. .........................................................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
..
.
.
.
.
.
.
.. . .
...............
. ......... ..................................................................... .......................... ..........................
. . .... ... ...... ... . . .... ...
.
.. .. .. .... .
..
-0.2
0
2
4
8 10
6
8 10
0.15
6
8 10
x/
0.1
d)
Bz
x
0.0
6
x/
c)
0.2
4
0.05
0.0
0
2
4
x/
Figure 11: PIC simulation of resonance absorption for parameters
θ = 9o , vos /c = 0.07, L/λ = 5 (kL = 10π), and nemax /nc = 1.5 : a)
Density profile and electric field normal to the target, b) electric field
parallel to the target, c) particle momenta, d) laser magnetic field.
Interaction with Solids
Collisionless Absorption
Resonance absorption
84 132
Vacuum heating: Brunel model
Resonance absorption not possible if oscillation amplitude
exceeds the density scale length L, i.e. if vos /ω > L.
Interaction with Solids
Collisionless Absorption
Brunel model
85 132
Vacuum heating: Brunel model
Resonance absorption not possible if oscillation amplitude
exceeds 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 the
target surface.
vd
Ed
EL
k
θ
111
000
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
000
111
0
∆x
ne
x
Figure 12: Capacitor model of the Brunel heating mechanism.
Interaction with Solids
Collisionless Absorption
Brunel model
85 132
Brunel model II
Driving electric field
Ed = 2EL sin θ.
(52)
Field pulls a sheet of electrons out to a distance ∆x from its
initial position. Surface number density of this sheet is
Σ = ne ∆x, so the electric field created between x = −∆x and
x = 0 is
∆E = 4π eΣ.
Equating this to the driving field and solve for Σ:
Σ=
Interaction with Solids
2EL sin θ
.
4π e
Collisionless Absorption
(53)
Brunel model
86 132
Brunel model III
When the charge sheet returns to its original position, it acquires
velocity vd ' 2vos sin θ, where vos is the usual electron quiver
velocity.
Assuming these electrons are all ‘lost’ to the solid, then the
average energy density absorbed per laser cycle is given by:
Pa
Interaction with Solids
=
Σ mvd2
τ 2
'
1
e 3
E .
2
16π mω d
Collisionless Absorption
Brunel model
87 132
Brunel model IV
Compare to the incoming laser power:
PL = cEL2 cos θ/8π.
Substituting Eq. (52), we obtain the
fractional absorption rate for the Brunel mechanism:
ηa ≡
Pa
4 sin3 θ
= a0
,
PL
π cos θ
(54)
where a0 = vos /c.
Expect more absorption at large angles of incidence and higher
laser irradiance, I λ2 ∝ a02 .
Interaction with Solids
Collisionless Absorption
Brunel model
88 132
Brunel model: refinements
Two corrections to improve the model:
1
Account for the reduced driver field amplitude due to
absorption, replacing Eq. (52) by
Ed = [1 + (1 − ηa )1/2 ]EL sin θ.
2
(55)
Relativistic return velocities: replace kinetic energy with
Uk = (γ − 1)mc 2
Interaction with Solids
Collisionless Absorption
Brunel model
89 132
Vacuum heating
Interaction with Solids
Collisionless Absorption
Brunel model
90 132
Fully relativistic model
Gibbon, Andreev & Platanov (2011)
1.1
1.0
a0=0.027
a0=0.085
a0=0.27
a0=0.85
a0=2.7
1.0
0.9
0.8
=0
=6
=45
=60
=85
0.9
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
17
10
0
10
20
30
40
50
(deg)
60
70
80
90
2
5
10
18
2
I
5
2
10
19
-2
[Wcm
2
5
10
20
2
5
21
10
2
m]
Figure 13: Absorption fraction vs. angle and intensity
Interaction with Solids
Collisionless Absorption
Brunel model
91 132
Hot electron generation
Typical signature of collisionless laser heating – bi-Maxwellian
electron spectrum with characteristic temperatures Tc and Th .
5
10
Te ~ 5 keV
4
f(U)dU
10
3
10
2
10
Th ~ 75 keV
10
1
0
100
200
300
400
500
U (keV)
Interaction with Solids
Hot Electron Generation
92 132
Short pulse Th scaling
Simple Brunel model:
ThB =
mvd2
' 3.7I16 λ2µ
2
(56)
EM PIC simulations:
ThGB ' 7 I16 λ2
1/3
,
(57)
Relativistic j × B model (ponderomotive scaling):
ThW
Interaction with Solids
' mc 2 (γ − 1)
h
i
1/2
' 511 1 + 0.73I18 λ2µ
− 1 keV.
Hot Electron Generation
Scaling
(58)
93 132
Hot electron temperature: short pulses
5
10
Thot (keV)
LLNL-00
Experiments
Th (FKL)
Th (W)
Th (GB)
Th (B)
2
3
RAL-99
IOQ-97
5
CELV-96
MBI-00
RAL-96
STA-92
LLNL-99
IOQ-00
2
2
CELV-96
MBI-95
10
MBI-97
5
LULI-97
2
5
INRS-99
IOQ-96
10
IOQ-99
LULI-94
LLE-93
2
15
10
16
17
10
I
18
10
10
2
-2
(Wcm
19
10
20
10
2
m)
Figure 14: Hot electron temperature measurements in femtosecond
laser-solid experiments (squares) compared with various models: long
pulse — Eq. (??); Brunel — Eq. (56); Gibbon & Bell — Eq. (57) and
Wilks — Eq. (58).
Interaction with Solids
Hot Electron Generation
Measurements of hot electron temperature
94 132
Ion acceleration
Direct acceleration in laser field inefficient, since
vi
eEL
mi
a0
'
=
a0 ≤
c
mi ω c
me
1836
Interaction with Solids
Ion acceleration
95 132
Ion acceleration
Direct acceleration in laser field inefficient, since
vi
eEL
mi
a0
'
=
a0 ≤
c
mi ω c
me
1836
Get relativistic ion energies for
a0
or
Interaction with Solids
I λ2L
∼ 2000
≥ 5 × 1024 Wcm−2 µm2
Ion acceleration
95 132
Ion acceleration
Direct acceleration in laser field inefficient, since
vi
eEL
mi
a0
'
=
a0 ≤
c
mi ω c
me
1836
Get relativistic ion energies for
a0
or
I λ2L
∼ 2000
≥ 5 × 1024 Wcm−2 µm2
Therefore need means of transmitting laser energy to ions over
many cycles → exploit electrostatic field in plasma.
Interaction with Solids
Ion acceleration
95 132
Mechanisms
1
Coulomb explosion: clusters; ponderomotive channelling in
gas jets
Interaction with Solids
Ion acceleration
Mechanisms
96 132
Mechanisms
1
Coulomb explosion: clusters; ponderomotive channelling in
gas jets
2
Electrostatic sheath formed by hot electron cloud (TNSA)
Interaction with Solids
Ion acceleration
Mechanisms
96 132
Mechanisms
1
Coulomb explosion: clusters; ponderomotive channelling in
gas jets
2
Electrostatic sheath formed by hot electron cloud (TNSA)
3
Collisionless shock formation: hole boring
Interaction with Solids
Ion acceleration
Mechanisms
96 132
Mechanisms
1
Coulomb explosion: clusters; ponderomotive channelling in
gas 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
96 132
Sheath model
Electrostatic 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
e
= (ne − Zni )
2
∂x
ε0
Interaction with Solids
Ion acceleration
Sheath model
97 132
Sheath model
Electrostatic 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
e
= (ne − Zni )
2
∂x
ε0
Ion expansion is described by continuity and momentum
equations:
∂
∂
+ vi
ni
∂t
∂x
∂
∂
mi
+ vi
vi
∂t
∂x
Interaction with Solids
Ion acceleration
∂ vi
∂x
∂φ
= −Ze
∂x
= −ni
Sheath model
(59)
97 132
Self-similar solution
If the plasma stays quasineutral everywhere (ne ' Zni ), then
Eqs. (59) have a self-similar solution in x /t:
Zni
vi
= ne0 exp(−x /cs t − 1)
= cs + x /t
(60)
x
+ 1)
eφ = −Te (
cs t
where cs =
Interaction with Solids
p
ZTe /mi is the ion sound speed.
Ion acceleration
Sheath model
98 132
Self-similar solution
If the plasma stays quasineutral everywhere (ne ' Zni ), then
Eqs. (59) have a self-similar solution in x /t:
Zni
vi
= ne0 exp(−x /cs t − 1)
= cs + x /t
(60)
x
+ 1)
eφ = −Te (
cs t
p
where cs = ZTe /mi is the ion sound speed.
Max ion velocity is
vf = 2cs log(τ +
p
τ 2 + 1)
where
ωpi t
τ = √ ; ωpi =
2e
Interaction with Solids
Ion acceleration
Z 2 e2 ni0
ε0 m i
(61)
1/2
Sheath model
98 132
Energy spectrum
Ion energy spectrum is given by:
dN
ni0 cs t
1/2
=
e−(2UU0 )
1
/
2
dU
(2UU0 )
(62)
where U0 = ZkB Th .
Figure 15: Ion energy spectrum from Mora expansion model
Interaction with Solids
Ion acceleration
Sheath model
99 132
Hole boring
On the front side of an overdense plasma, the ponderomotive
force will displace and compress electrons into the target,
creating an electrostatic field acting on the ions.
Interaction with Solids
Ion acceleration
Hole boring
100 132
Hole boring
On the front side of an overdense plasma, the ponderomotive
force will displace and compress electrons into the target,
creating an electrostatic field acting on the ions.
Momentum balance across the collisionless, electrostatic shock
thus formed implies:
ui = 2us = 2
IL
mi ni c
s
1/2
=2
Z me n c
A mp ne
(63)
where us is the velocity of the shock front.
Interaction with Solids
Ion acceleration
Hole boring
100 132
Hole boring
On the front side of an overdense plasma, the ponderomotive
force will displace and compress electrons into the target,
creating an electrostatic field acting on the ions.
Momentum balance across the collisionless, electrostatic shock
thus formed implies:
ui = 2us = 2
IL
mi ni c
1/2
s
=2
Z me n c
A mp ne
(63)
where us is the velocity of the shock front. This leads to a
quasi-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
100 132
Light sail acceleration
A mass-limited target, such as a nm-thick foil, allows nearly
complete displacement of electrons, thereby maximizing the ES
field. A simple capacitor model suffices to determine the
threshold intensity for this scenario.
Interaction with Solids
Ion acceleration
Light sail
101 132
Light sail acceleration
A mass-limited target, such as a nm-thick foil, allows nearly
complete displacement of electrons, thereby maximizing the ES
field. A simple capacitor model suffices to determine the
threshold intensity for this scenario.
The charge separation field of a foil with thickness d is:
∆E =
e
ε0
ne d
This is balanced by the net laser field at the surface,
2EL = 2me ω ca0 /e, leading to the matching condition:
a0 ' π
Interaction with Solids
Ion acceleration
ne d
nc λL
(64)
Light sail
101 132
Light sail acceleration
A mass-limited target, such as a nm-thick foil, allows nearly
complete displacement of electrons, thereby maximizing the ES
field. A simple capacitor model suffices to determine the
threshold intensity for this scenario.
The charge separation field of a foil with thickness d is:
∆E =
e
ε0
ne d
This is balanced by the net laser field at the surface,
2EL = 2me ω ca0 /e, leading to the matching condition:
a0 ' π
ne d
nc λL
(64)
Under these conditions, find max ion energy Ui ∼ t 1/3 – see
Esirkepov, PRL (2004).
Interaction with Solids
Ion acceleration
Light sail
101 132
Summary of Lecture 3
Short pulse interaction scenarios
Collisional Absorption
Collisionless Absorption
Hot Electron Generation
Ion acceleration
Interaction with Solids
Ion acceleration
Light sail
102 132
Mitglied der Helmholtz-Gemeinschaft
Physics of High Intensity
Laser Plasma Interactions
Part IV: Numerical Simulation of
Laser-Plasma Interactions
20–25 June 2011
Paul Gibbon
Lecture 4: Numerical Simulation of Laser-Plasma
Interactions
Plasma models
Classification
Hierarchy
Hydrodynamics
Thermal transport
Single fluid model
Numerical scheme
Particle-in-Cell Codes
Vlasov equation
Particle pusher
Field solver
Algorithm
3D codes
Numerical Simulation of Laser-Plasma Interactions
104 132
-3
Electron density (cm )
Plasma classification
10
32
10
28
10
24
10
18
10
14
10
10
10
6
10
2
white dwarf
l
ea
id
nno
as
sm
pla
stellar interior
Jovian interior
electron gas
in a metal
laser-plasmas
solar
atmosphere
magnetic
fusion
solar corona
ionosphere
(F layer) gaseous nebula
10
4
10
Temperature (K)
10
7
10
10
Figure 16: Plasma classification in the density-temperature plane.
Numerical Simulation of Laser-Plasma Interactions
Plasma models
Classification
105 132
Plasma model hierarchy
non−eqm
Molecular dynamics
Microscopic
equilib
Monte−Carlo
rium
statistical
average
Kinetic theory:
Boltzmann equation
collisionless
Vlasov
Particle−in−Cell
coll
is
ion
al
velocity
moments
Fokker−Planck
Hydrodynamics
Macroscopic:
fluid equations
+ E, B
MHD
d/dt = 0
Steady−state
Figure 17: Physical basis of common plasma models and
corresponding numerical approaches.
Numerical Simulation of Laser-Plasma Interactions
Plasma models
Hierarchy
106 132
Numerical modelling of laser-plasma interactions
Two types of modelling dominate LPI:
Numerical Simulation of Laser-Plasma Interactions
Plasma models
Hierarchy
107 132
Numerical modelling of laser-plasma interactions
Two types of modelling dominate LPI:
1
Hydrodynamic modelling to follow the macroscopic, dynamic
behavior of the plasma, including external electric and
magnetic fields, and heating by laser or particle beams: ps-ns
timescale. Good for prepulse modelling.
Numerical Simulation of Laser-Plasma Interactions
Plasma models
Hierarchy
107 132
Numerical modelling of laser-plasma interactions
Two types of modelling dominate LPI:
1
Hydrodynamic modelling to follow the macroscopic, dynamic
behavior of the plasma, including external electric and
magnetic fields, and heating by laser or particle beams: ps-ns
timescale. Good for prepulse modelling.
2
Kinetic modelling for situations in which ’non-Maxwellian’
particle distributions fα (v ) have to be determined
self-consistently – this is the method of choice for
laser-particle accelerator schemes!
Numerical Simulation of Laser-Plasma Interactions
Plasma models
Hierarchy
107 132
Hydrodynamic plasma simulation
Figure 18: Building blocks of a laser-plasma hydro-code.
Numerical Simulation of Laser-Plasma Interactions
Hydrodynamics
108 132
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.
Numerical Simulation of Laser-Plasma Interactions
Hydrodynamics
Thermal transport
109 132
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 ' ls πσL2 .
Numerical Simulation of Laser-Plasma Interactions
Hydrodynamics
Thermal transport
109 132
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 ' ls πσL2 .
Setting = 23 ne kB Te and ∇.Φa ∼ Φa /ls , have
dTe
Φa
'
,
dt
ne ls
(65)
heating rate:
d
Φa
(kB Te ) ' 4
dt
Wcm−2
Numerical Simulation of Laser-Plasma Interactions
ne
cm−3
−1 Hydrodynamics
ls
cm
−1
keV fs−1 .
Thermal transport
(66)
109 132
Thermal transport
Energy transport equation for a collisional plasma:
∂
+ ∇.(q + Φa ) = 0,
∂t
(67)
where is the energy density, q is the heat flow and Φa = ηa ΦL
is the absorbed laser flux.
Numerical Simulation of Laser-Plasma Interactions
Hydrodynamics
Thermal transport
110 132
Thermal transport
Energy transport equation for a collisional plasma:
∂
+ ∇.(q + Φa ) = 0,
∂t
(67)
where is the energy density, q is the heat flow and Φa = ηa ΦL
is the absorbed laser flux.
Huge temperature gradients are generated after a few fs: heat is
carried away from the surface into the colder target material
according to Eq. (67). For ideal plasmas, we write
=
3
ne kB Te
2
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
110 132
Spitzer-Härm heat-flow
Substituting for and q in Eq. (67) and restricting ourselves to
1D by letting ∇ = (∂/∂ x , 0, 0), gives a diffusion equation for Te :
3
∂ Te
∂
ne kB
=
2
∂t
∂x
∂ Te
κe
∂x
+
∂ΦL
.
∂x
(69)
κe is known as the Spitzer thermal conductivity and is given by:
1/2
κe = 32
where
ν0 =
Numerical Simulation of Laser-Plasma Interactions
2
π
ne
5/2
Te ,
5
/
2
ν0 m
(70)
2π ne Ze4 log Λ
.
m2
Hydrodynamics
Thermal transport
111 132
Nonlinear heat wave
400
t = 100 fs
t = 200 fs
t = 300 fs
t = 400 fs
Te (eV)
300
200
100
0
0
40
80
120
160
200
x (nm)
Figure 19: Nonlinear heat-wave advancing into a semi-infinite,
solid-density plasma. The curves are obtained from the numerical
solution of the Spitzer heat flow equation for constant laser absorption
at the target surface (left boundary).
Numerical Simulation of Laser-Plasma Interactions
Hydrodynamics
Thermal transport
112 132
Single-fluid approximation
Fluid model
∂ρ
+ ∇·(ρu ) = 0 (71)
∂t
∂ρu
+ ∇·(ρuu ) + ∇P − f p = 0 (72)
∂t
Q ei
∂e
+ ∇· u (e + Pe ) − κe ∇Te −
− Φa = 0 (73)
∂t
γe − 1
∂i
Q ei
+ ∇· u (i + Pi ) − κi ∇Ti +
= 0,(74)
∂t
γi − 1
Numerical Simulation of Laser-Plasma Interactions
Hydrodynamics
Single fluid model
113 132
Notes
Average fluid density ρ and velocity u defined thus:
ρ ≡ ni M + ne m ' ni M ,
u
≡
1
ρ
(ni Mu i + ne mu e ) ' u i .
The energy density α is the sum of internal and kinetic fluid
energies:
1
Pα
+ ρu 2 ,
α =
γα − 1 2
with γα defined as the number of degrees of freedom.
Qei is the electron-ion equipartition rate
Qei =
2m ne kB (Te − Ti )
M
τei
(75)
τei = νei−1 is the electron-ion collision time.
Coupling to the laser Φa and f p .
Numerical Simulation of Laser-Plasma Interactions
Hydrodynamics
Single fluid model
114 132
Example: laser prepulse heating solid Al target
Figure 20: Density (solid line) and temperature (dashed line) profiles
from 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, 300
fs 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
115 132
Kinetic plasma simulation
Simplest kinetic description of a plasma uses single-particle
velocity distribution function f (r , v ), evolving according to the
Vlasov equation:
∂f
v
∂f
∂f
+ v·
+ q (E + × B )·
= 0.
∂t
∂x
c
∂p
(76)
Distribution function f (r , v ) is 6-dimensional – direct solution
of Eq. (76) generally impractical.
Even 1D spatial geometry still needs 2 or 3 velocity
components to couple to Maxwell’s equations, ie: 3- or
4-dimensional code.
Numerical Simulation of Laser-Plasma Interactions
Particle-in-Cell Codes
Vlasov equation
116 132
Particle-in-Cell simulation
Important method developed in the 1960s is the so-called
Particle-in-Cell (PIC) technique.
Distribution function is represented instead by a large number of
discrete macro-particles, each carrying a fixed charge qi and
mass mi .
q(x ,p )
p
i
f(x,p)
i
p
x
x
Figure 21: Relationship between a) Vlasov and b) PIC approaches.
Numerical Simulation of Laser-Plasma Interactions
Particle-in-Cell Codes
Vlasov equation
117 132
Particle pusher
Geometry for simplified Lorentz equation (see Eq. 16):
E = (0, Ey , 0), B = (0, 0, Bz ):
1/2-acceleration:
n− 21
px− = px
n− 21
; py− = py
+
∆t
2
Eyn
rotation:
∆t Bzn
2 γn
−
= px + py− t
γ n = (1 + (px− )2 + (py− )2 )1/2
;
px0
py+
px+
t=
;
s=
2t
1 + t2
= py− − px0 s
=
px0
+
(77)
px+ t
1/2-acceleration:
n+ 12
px
n+ 12
= px+ ; py
Numerical Simulation of Laser-Plasma Interactions
= py+ +
Particle-in-Cell Codes
∆t
2
Eyn .
Particle pusher
118 132
Density & current gather
The density and current sources needed to integrate Maxwell’s
equations are obtained by mapping the local particle positions
and velocities onto a grid as follows:
ρ(r ) =
X
qj S (rj − r ),
j
J (r ) =
X
qj v j S (rj − r ),
j = 1...Ncell
(78)
j
where S (rj − r ) is a function describing the effective shape of
the particles. Usually it is sufficient to use a linear weighting for
S — the ‘Cloud-in-Cell’ scheme — although other more accurate
methods (eg: cubic spline) are also possible.
Numerical Simulation of Laser-Plasma Interactions
Particle-in-Cell Codes
Field solver
119 132
Field solver: FDTD
Once ρ(r ) and J (r ) are defined at the grid points, we can
proceed to solve Maxwell’s equations to obtain the new electric
and magnetic fields. In 2D geometry, can define:
F+
=
F−
=
1
(Ey + Bz )
2
1
(Ey − Bz )
2
Then Faraday and Ampere laws reduce to (normalised units):
Fi+
+1 (n + 1) = Fi (n) −
∆t
2
Fi− (n + 1) = Fi +1 (n) −
Numerical Simulation of Laser-Plasma Interactions
n+1/2
Jy ,i +1/2 (forward)
∆t
2
n+1/2
Jy ,i +1/2 (backward) (79)
Particle-in-Cell Codes
Field solver
120 132
PIC algorithm summary
The fields are then interpolated back to the particle positions so
that we can go back to the particle push step Eq. (??) and
complete the cycle, see Fig. 22.
4. Push particles
ri vi
Ei Bi
3. Mesh −> Particles
1. Particles −> Mesh
ρg jg
Eg Bg
2. Maxwell equations
Figure 22: Iteration steps of the particle-in-cell algorithm.
Numerical Simulation of Laser-Plasma Interactions
Particle-in-Cell Codes
Algorithm
121 132
PIC codes for laser-plasma interaction studies
Because of its simplicity and ease of implementation, the
PIC-scheme sketched above is currently one of the most
important plasma simulation methods.
Name
Authors
Group(s)
OSIRIS
VLPL
REMP
VPIC
EPOCH
CALDER
SPSC
Mori, Silva
Pukhov
Esirkepov
Lin, Bowers
Arber, Bennett
Lefebvre
Ruhl, Brömmel
UCLA, IST Lisbon
U. Düsseldorf
Kyoto, Moscow
LANL
U. Warwick (CCPP)
CEA
LMU, FZ Jülich
Table 3: Parallel 3D, electromagnetic PIC codes
Numerical Simulation of Laser-Plasma Interactions
Particle-in-Cell Codes
3D codes
122 132
Summary of Lecture 4
Plasma models
Hydrodynamics
Particle-in-Cell Codes
Numerical Simulation of Laser-Plasma Interactions
Particle-in-Cell Codes
3D codes
123 132
Mitglied der Helmholtz-Gemeinschaft
Physics of High Intensity
Laser Plasma Interactions
Part V: Tutorial on Particle-in-Cell
Simulation
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. RPA
Tutorial on Particle-in-Cell Simulation
125 132
The PIC code BOPS
BOPS is a one-and-three-halves (1 spatial, 3 velocity
coordinates: 1D3V) particle-in-cell code originally created by
Paul Gibbon and Tony Bell in the Plasma Physics Group of
Imperial College, London.
Based on standard algorithm for a 1D electromagnetic PIC
code from Birdsall & Langdon.
Optionally employs a Lorentz ‘boost’ along the target surface
to mimic 2D, periodic-in-y geometry, with big savings in
compute time.
Applications: absorption, electron heating, high harmonic
generation, ion acceleration
Tutorial on Particle-in-Cell Simulation
The PIC code BOPS
126 132
Prerequisites
Current version: BOPS 3.4
Download site:
https://trac.version.fz-juelich.de/bops
Compiler: sequential code written in Fortran 90 (Intel ifort or
GNU’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
127 132
Windows users
1
First install CYGWIN (www.cygwin.com). This creates a
fully-fledged Unix environment emulator under Windows. In
addition to the default tools/packages offered during the
installation 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 under
cygwin (eg using gnuplot)
2
If you forget anything first time, just click on the Cygwin
installer icon to locate/update extra packages.
Download and unpack the bops3.4.tar.gz file with an archiving
tool (eg PowerArchiver: www.powerarchiver.com). Put this in
your ’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
128 132
Installation
The directory structure resulting from unpacking the tar files
should look like this:
src/
... containing the fortran90 source code
doc/
... some documentation in html and ps
tutorial/
... scripts and files for current tutorial
benchmarks/ ... additional sample scripts for running the code
tools/
... postprocessing tools
To compile:
Go to the source directory src, and edit the Makefile: adjust
and tune the flags to match your machine type (FC=ifort or
gfortran etc). Then do:
make
Tutorial on Particle-in-Cell Simulation
Installation
129 132
Running BOPS
Once the code has compiled, go back up to the base (or top)
directory and enter the tutorial directory. Here you will find the
run 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 in
wake.sh first:
tutorial/wake.sh
Further examples can be found in benchmarks
Tutorial on Particle-in-Cell Simulation
Running BOPS
130 132
Project I: Laser wakefield accelerator
1
Edit and run the script: ./wake.sh. Note that the input
parameters are normalised to laser wavelength and
wavenumber.
3
Examine the field and particle phase space plots at
successive time intervals (e.g.: t=100, t=200). The
gnuplot script wake.gp should help you get started.
2
Look at the printed output to check the actual laser and
plasma parameters (are they what you intended?)
4
How can the plasma wave amplitude | Ex |max be
optimised? Hint: try changing the electron density and/or
pulse length.
Plot files
5
Increase the laser amplitude to (a0 = 3) and observe the
difference in the electron phase space and plasma wave.
How does the matching condition need to be altered in this
regime? Why is the amplitude of the latter reduced towards
the back of the wake?
6
(advanced) Set up a plasma equivalent to twice the
dephasing length (Eq. 37) and determine the maximum
energy of electrons trapped in the plasma wave. Can you
beat the scaling predicted by Eq. 38?
These are in run directory (foil_tnsa1) in ASCII format
and have a suffix NN.xy, where NN is the snapshot
number. See bops.oddata for complete list
ezsi
nenc
exsi
pxxe
EM field Ez (S-polarized wave)
Electron density
Electrostatic field
Electron momenta (x − px phase space)
Project II: Ion acceleration – TNSA vs. RPA
1
TNSA regime.
a) The script foil.sh sets up a 2 µm ‘foil’ out of frozen
hydrogen (Z = A = 1), with density ne /nc = 36. This is
irradiated by a linearly polarized 50 fs pulse with intensity
1019 Wcm−2 .
Run the script and inspect the following plots at the
snapshot times (0, 50, 100, 150) fs:
Plot files
nenc,
exdc
pxxe,
fuep,
uinc,
ninc Electron & Ion densities
Cycle-averaged electrostatic field
pxxi Electron & ion momenta
fuip Electron, ion energy spectra
ubac Incoming and outgoing wave energy
b) Try varying, eg: pulse amplitude or duration and
compare the scaling of the maximum ion energy against
the theoretical prediction of Eq. (15).
2
HB regime.
a) Copy the script to a new file (eg: hb.sh), switch the
polarization from linear to circular (set cpolzn='C'),
change the run directory – eg: RUN=hb1 and rerun. (What
phase space variables can you check the laser polarization
with?) 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 in
this case is to avoid numerical heating arising from an
underresolved (cold) electron Debye length. Adjust the grid
size/resolution to ensure that ∆x < 0.5λD . Rerun and
compare the ion shock velocity with Eq. (63).
3
Light sail regime.
a) Use the matching condition Eq. (64) to determine the foil
width (using the same laser and target material) for which
the light sail regime is reached. Set ilas=4 with
tpulse=50, tfall=5, and ensure that the new foil is
resolved by the grid. Create a new script with these
parameters and rerun. NB: you may have to increase the
parameter uimax for this case!
b) Investigate the scaling behaviour of the monoenergetic
ion feature with intensity, pulse length etc.
c) Now repeat with a Gaussian or sin2 pulse shape.
Compare the hot electron spectrum and phase space with
the flat-top case at early times: what causes this
difference? What could you change to improve the far-field
ion sail stability for realistic pulse forms?
d) Now try the same thing with a carbon foil. Here, you will
need higher (solid) number densities and multiply charged
ions (see Eq. 41). What happens to the max. ion energy?
How does the energy/nucleon compare with the hydrogen
case?