Simulations of capillary
discharges
Prague, November, 2012
INTRODUCTION
• Owing to an extreme simple construction of the capillary
discharges and to a broad range of their applications the
capillaries have attracted a great deal of attention.
• The main constitutive element of the capillary is a channel
made in an insulator material and ended in the longitudinal
direction by the two electrodes. An external low-inductance
electric circuit produces the voltage between the electrodes
and generates the electric current pulse in the channel.
• The current heats the wall material and generates the
magnetic field. As a result the prefilled capillary plasma and
the plasma ablated from the wall is pinched causing the
shock wave to propagate towards the axis and to form
there the dense, high temperature short living filament.
• The plasma filament parameters can correspond to the
conditions for the lasing in the XUV and X-ray region. This
is connected with one of the main applications of the
capillary discharges.
Another important application of the capillary discharges is
related to their property to form on the long time scale evolution
inside the capillary the plasma density profile with a density
minimum at the axis which provides the ultra-short laser pulse
guiding (Ehrlich, et al., 1996). In this case the capillary plays a role
an optical waveguide for a laser pulse (Tajima, 1985) providing its
propagation over a distance greater than the defocusing length.
The interaction of intense laser radiation with plasmas is important
in a wide variety of applications such as the generation of coherent
short-wavelength radiation through high-harmonic generation ,
X-ray lasers, gamma ray generation, and the development of novel
plasma-based accelerators.
For such applications the laser-plasma interaction length is
limited fundamentally by diffraction to lengths of the order of the
Rayleigh range
ZR   w20 / 
where w0
is the laser waist size.
In order to increase the distance over which the intensity of the
laser pulse is maintained at a value close to that at its focus, it is
necessary to channel the laser pulse in some manner.
Table-top soft x-ray lasers
Electric current pulse
Schematic view of setup
Pin-hole images of the soft-x-ray
emitting region
Simulation
On axis spectra of Cd capillary
discharge plasma emission from 13.2
nm line of Ni-like Cd
J. J. Rocca, Rev. Sci. Instr. 70, 3799 (1999)
• A number of important applications, such as shortwavelength lasers and novel schemes for particle
acceleration, involve the interaction with plasmas of
ultra-short laser pulses with peak intensities in the
range
1624
2
10
W / cm
• The laser-plasma interaction length is limited by
diffraction to distances of order the Rayleigh range.
The propagation of an intense laser pulse through a
partially ionized plasma can also be limited by
refrection.
In order to increase the distance over which the
intensity of the laser pulse is maintained at a value
close to that at its focus, it is necessary to channel
the laser pulse in some manner. For example, it can
be guided inside the initially performed hollow or
prefilled with plasma narrow channel.
1.0 GeV Beam Generation in Laser Interaction with Capillary
Plasma
Laser: 1.5 J/pulse
Density: 4x1018 cm-3
Capillary: 312 mm diameter and 33 mm length
1 GeV beam: a0 ~ 1.46 (40 TW, 37 fs)
CAPILLARY
Peak energy: 1000 MeV
Divergence(rms): 2.0 mrad
Energy spread (rms): 2.5%
Charge: > 30.0 pC
W.P.Leemans et al, Nature Physics, 418 (2006)
Capillary discharges are an attractive method for forming a plasma
waveguide.
•In fast z-pinch discharges the very fast rising current causes the
plasma to be pinched, driving a strong shock wave towards the axis.
The collapsing annulus of highly ionized plasma can form a transient
plasma channel. The channel may be formed in either initiallyevacuated capillaries or gas filled capillaries. Hosokai et al. Opt.Lett.,
2000, Fauser and Langhoff, 2000.
•The major disadvantage of this approach is that the plasma channel
exists for only a few nanoseconds which places severe restrictions on
the allowable timing jitter between the start of the capillary discharge
and the arrival of the laser pulse.
Physical model
We use the approximation of two-temperature, one-fluid MHD.
Owing to the large length-to-radius ratio of the capillary
l / R0  1
a one-dimensional approximation is considered .
The following phenomena are essential for capillary discharge in
evacuated channel:
the interaction of the plasma with a wall, and the ablation of the wall
material accompanied by the ionization of the produced gas
the interaction of the magnetic field with the
plasma (pinch effect).
Important dissipative processes:
electron and ion thermal conductivities,
Nernst and Ettinghausen effects,
Joule heating,
radiation losses, viscosity.
It is also necessary to incorporate the degree of ionization both
into the equation of state and into the dissipative coefficients.



0
z 
j   0, 0, j z 
B   0, B , 0 
v   v, 0, 0 
B
Ez
t
r
Rz vB
Ez 

,
ene
c
c
 (e)
 ( e )
Te vB
Ez  
jz 
B

ene
ene
r
c

B

Te 

  c2 1 
 

vB  
(rB )   c
N B




t
r
r  4  r r

r

r



System of equations
 1 

r v  0
t r  r
  v v   p 1

1
   v     jB   rr    rr   
r
r
 t r   r c
 B  B c 
E
 v   
t   r  r   r   r r
 
  p 
1 
  e  v e   e  r v   jE 
 r qe   Qr  Cei Ti  Te 
 r  r r
r r
 t
 
  p 
1 
1
v
  i  v i   i  r v   jE 
r
q

C
T

T


r
v

 i  ei  e i  rr       rr 
r  r r
r r
r r
r
 t
Dissipative coefficients
 rr
E
v
2
v 
 0   2
,
3
r 
r
j

N B
qi   i 
 Ti
r
 Te
,
r
1

qe   e 
 Te
 N BTe j ,
r
neTe
1  xe , w ,
me e
N
me ei
 2 1  5  xe , w  ,
e ne
me
Cei  3
ne ei ,
AmA
 e 

eB
xe 
,
me c ei

2
2 v
 0 r
3
 r2
w
 ee
2 z2  ei
1
4  xe , w ,
mec ei
0  0.96 ni Ti  ii1
i xe , w 
i ,1 wxe2  i , 2 w
xe4
 7,1 w
xe2
 7, 2 w
2
,
i , j w  i3, j w3  i2, j w2  i1, j w  i0, j ,
i  1,2,,6,
i  1,2,,7;
j  1,2.
Expressions for i, j are obtained in the so called two-polynominal approximation
i xe , w are determined with the accuracy of a few percent, which is quite sufficient
because the accuracy of the Landau approximation for the collision integral
is of the order of 1 /  .
l
The values of i xe , w
multiples of
i, j
 2w 
 2  and
1

ei
i  1,,7 
are calculated by Braginskii for for certain
which is not very convenient when w runs through a
continuous set of values. For example, values of
i, j 1/ 2

differ by more than a factor of 2.
We take into account the possible considerable difference between  ee and  ek
otherwise the accuracy of the resulting system of equations is incorrectly reduced.
Neutral particles
4 2 e4 ne z2 ei
 ei 
,
3
3
me Te 2
4  e 4 ni z14 ii
 ii 
,
3
3
AmA Te 2
i
z1  z для z  1, z1  1 для z  1
z2 ei  ei  en для z  1
z2 ei  zei
en 
для z  1
1 z
4 z 1   0 / Te  


2
,
4
m
e
4 e
o 
3 4
Equation of state and degree of ionization
z 6
For the equation of state and the ionization degree, the
approximation of local thermodynamic equilibrium is used
separately for the electron and ion components.
  z    z  0
  z
  z
- is the chemical potential of an ideal free-electron gas
-the ionization potential of a mean ion.
1<z<Z/2 -- Sommerfeld's formula in the Thomas-Fermi model for the ion
shell
Z/2<z<Z -- the formula for the hydrogen-like ionization potential is chosen,
taking into account thescreening of the ion electric field by $(Z-z-1)$ electrons
z0  z  1 the simplified Saha formula is used, taking only neutral and once
ionized atoms .
z0  106
Thermodynamic properties
3/ 2

zTe
 meTe  2V  U  z 
Fe V , Te   
1  ln 


2
AmA 
z  AmA
 2 
V – удельный объем одного иона
  Fe 

 V , Te z V , Te    0
  z V ,Te

  Fe 
1 
 Fe V , Te   Te 
 e   , Te  
 V , Te  
A mA 

  Te V ,

  Fe 
pe   
 V , Te 
 V T
e
3
2
 i  Ti ,
pi 
Ti
V
Boundary conditions
r  Rex
dRex  t 
dt
Free boundary
 v  Rex  t  , t 

v
v
p

p




 pex  t 
 e

i
0
0

r
r

 Rex
c
B 
 e Te

0
   r  4 N Te r r  rB  

 Rex
или
Te  Rex  t  , t   Tex  t 
 Ti
0
r
B  Rex  t  , t  
r0
2I t 
cRex  t 
 Te
 Ti
v  r  0   0, B  r  0   0,
 0  qe  r  0   0  ,
0
t
t
Различные типы динамики капиллярного разряда
(физические процессы, которые определяют тип
динамики плазмы)
Динамика плазмы в капиллярных разрядах зависит от
нескольких параметров:
1.
2.
3.
4.
Радиуса капилляра
Параметров внешней электрической цепи
Материала стенок
В случае заполненного капилляра – начального давления
заполняющего капилляр газа, его атомного номера и веса
1. Роль магнитного поля:
а) основную роль играет пинч-эффект
б)влиянием магнитного поля можно пренебречь
2. Испарение стенок капилляра
Пинчевой капиллярный разряд
R0  2 мм
Радиус
Начальная плотность Ar
Капилляр из полиацетата
0  1.37 106 г / cм3
Z  7, A  14, 0  1г / cм3
Электрический ток
I (t )  I 0 sin( t / t0 ), I 0  40kA, t0  60нс
Столкновительное возбуждение неоноподобных ионов аргона
ne
Рокка и др. 1994
0.3 1.0 1019 см3 ,
Te
60  80эВ
Моделирование пинчевого капиллярного разряда
19.3
18.8
1
17.9
17.1
0
0
10
20
30
40
50
60
65
52
39
1
26
5E+19
10
20
30
40
50
60
40
30
1
20
10
0
0
0
10
20
30
time, ns
40
50
60
Ne , cm-3
2
0
0
80
4E+19
13
0
radius, mm
tc  R0 / vA , vA  B /  4 0 
1/ 2
Te , eV
radius, mm
2
10
60
3E+19
40
2E+19
20
1E+19
0
0
0
10
20
30
time, ns
40
50
60
Te , eV
19.7
electric current
kA
radius, mm
2
Log10 of electr.
density in cm-3
t> 20нс 30-40% I около стенки
<50% I по аргоновой плазме
5% I по керну
Comparison of the simulated trajectories of plasma elements
with the experimentally observed radius of radiative plasma
Radial distribution of the plasma parameters at t=40 ns
1.The increase of
on the axis.
 , Te , p, j
2. I is separated in two spatially
distinct components: near the
axis,
on the perepheiry of the
discharge.
3.The plasma density drop
corresponds to the boundary
between Ar and ablated plasma
Radial distribution of plasma parameters in the kernel
ne  4 1019 cm3
Te  60 eV
Trajectories of elements of argon plasma and
of ionized ablated material for different
Conclusion
• Kernel, close to the axis is likely to be the place,
where the amplification can take place. It is
uniformly filled with hot dense plasma.
• It is not pinch effect due to the lack of electric
current in the center of the channel.
• There is no MHD instabilities typical to Z-pinches
So the plasma behavior in the kernel can be
described in the frame of hydrodynamics.
• Electric current distribution is nonuniform. A
fraction, flowing near the wall causes heating and
ablation of the wall material. Another fraction,
localized near the axis, causes plasma acceleration
and compression at the initial stage of discharge.
•
New type of the capillary discharge waveguide in the
gas-filled channel was investigated by Spence and
Hooker, Phys.Rev. E, 2000. The current pulse had a
peak of 250 A and a duration of order 200 ns. A
hydrogen-filled capillary waveguide was used to guide
laser pulses with peak intensities of greater than
. 1016 W / cm2 through 20- and 40-mm long capillaries
with pulse energy transmitions of 92% and 82%,
respectively.
• It is important to understand the mechanism by which
the guiding electron density profile is formed. We
performed MHD simulations of the plasma dynamics
of a hydrogen-filled capillary discharge. The
mechanism of formation of the guiding electron
density profile is found to be very different from that of
Z-pinch capillary discharge.
Hydrogen capillary (nonpinching discharge)
Alumina capillary filled with hydrogen
R0  150m, I 0  250 A, t0  200ns, p0  67mbar
Three stages of the plasma evolution
1. Magnetic field penetrates the plasma on
a time scale of 1ns. Pinching of the plasma is
p  H 2 / 8
negligibly weak.
The plasma is heated and ionized locally.
Radial distributions of plasma parameters
are homogeneous. 50ns
2. A redistribution of the plasma temperature
and density occurs during t=50-80ns. Thermal
conduction becomes significant.
3. Quasi-state equilibrium at a given electric
current
t  5ns, t  10ns, t  2ns
h


The plasma temperature has its maximum on the axis, because the Ohmic
heating is balanced mainly by thermal conduction to cold wall. This results in
an axial minimum in the electron density profile.
Temporal evolution of the axial electron
temperature and density
3
10
ne
2
6
Te
4
1
Te , eV
ne , 1018 cm-3
8
2
0
0
50
100
t, ns
Experimental value
0
150
After t=80ns the axial electron
density is constant. The axial
electron temperature slowly
decreases with time.
At t=60ns
ne  2.8  1018 cm-3 , Te  3.4eV
ne  2.7  1018 cm-3
The measured and calculated electron
density profiles at t=60ns
ne, 1018 cm-3
4.5
4
t=55 ns
3.5
t=60 ns
t=65 ns
3
2.5
2
1.5
-150
-100
-50
0
50
100
150
d,m
Measured profile corresponds to an average electron density profile over several ns.
Allowing for errors in measurement of the time t of approximately 5 ns, and
averaging over the duration of the probe pulse used in the measurements, the
simulations are in good agreement with the measurements.
For practical applications of H-filled capillary discharge
waveguides, it is important that the capillary has a long lifetime.
The thermal flux due to
electron thermal conduction
Partial ionization of the wall and
heating of the free electrons
The lattice heating by energy
exchange with free electrons
Direct heating of the lattice due to ion-lattice thermal conduction plays
only a secondary role.
4
For t=100-150ns, the degree of ionization of atoms in the wall material  10
in a layer  0.8 m depth.
30 C at t = 50ns
no melting or

Lattice
Te  18
. eV
200 C at t = 100ns
disruption
temperature =
500 C at t = 150ns
In experiment after 10 discharge pulses the increase in capillary diameter was  1m.
5
Simple model of plasma equilibrium
For t>80 ns the following conditions are fulfilled:
• There is no screening of the axial electric field, and
consequently it is uniform across the capillary
• The magnetic field pressure is much less than the plasma
pressure, and hence the plasma pressure can be considered
to be constant across the capillary
• The electrons are unmagnetized xe   Be /  ei  0.04
• There is no difference between the electron Te and ion Ti
temperatures
The equation for heat flow
dT
dr
r 0
 0, T r  0  T 
1d
dT 
2
rκ

σ
E
0
 e


r dr 
dr 
boundary conditions
Assuming Coulomb logarithms to be constant,
 e   0T
5/ 2
    0T 3/ 2
Assuming that
electron thermal conductivity
electric conductivity
 e T r 0   e r  R T

we consider
0
Introducing a new variable
  r / R0 and new function u( ) determined by

T( )  7 0R E / 2 0
we obtain
T  0
2 2
0

1/ 2
u 2 /7
1 d  du 
du
3/ 7
 0, u  1  0
    u ,
 d  d 
d  0
The unique nontrivial solution of this equation is found numerically.
Assuming Coulomb logarithms to be constant,
 e   0T
5/ 2
    0T 3/ 2
Assuming that
electron thermal conductivity
electric conductivity
 e T r 0   e r  R T

we consider
0
Introducing a new variable
  r / R0 and new function u( ) determined by

T( )  7 0R E / 2 0
we obtain
T  0
2 2
0

1/ 2
u 2 /7
1 d  du 
du
3/ 7
 0, u  1  0
    u ,
 d  d 
d  0
The unique nontrivial solution of this equation is found numerically.
Assuming Coulomb logarithms to be constant,
 e   0T
5/ 2
    0T 3/ 2
Assuming that
electron thermal conductivity
electric conductivity
 e T r 0   e r  R T

we consider
0
Introducing a new variable
  r / R0 and new function u( ) determined by

T( )  7 0R E / 2 0
we obtain
T  0
2 2
0

1/ 2
u 2 /7
1 d  du 
du
3/ 7
 0, u  1  0
    u ,
 d  d 
d  0
The unique nontrivial solution of this equation is found numerically.
0.07
u(0)  0.067, u' 1  0.107
0.06
0.05
1
u
0.04
m0   u2/7  d  1.55

0.03
0.02
0
0.01
0.00
0
0.2
0.4
0.6
0.8
1

The pressure is constant across the capillary
p  2ne r Tr   2ne 0T0
 u0  

The electron density profile ne r   ne 0 
 ur  
The axial electron density
2/7


r2
 ne 0 1  0.33 2  
R0


ne 0 
1

 0.7364
2/7
ne
2m0u0 
t=100ns
8
6
Te
Te , eV
6
4
5
ne
4
simulation
5
3
3
2
2
ne , 1018 cm-3
7
simple model
1
1
0
0
50
100
150
0
200
r, m
The equilibrium state of the capillary discharge depends only on the electric current,
the capillary radius, the total mass of hydrogen per unit length of the capillary.
The time scale of outflow
  150  250ns
for 3-5mm long capillaries
Laser pulse guiding inside the
capillary
A Gaussian beam will propagate through a plasma channel with a parabolic density
profile of the form
n (r)  n (0)  n'' (0)r 2 / 2 with a constant spot size
e
e
e
1/ 4


2

WM  
"
  re ne (0) 
From the equilibrium model
re
- the classical electron radius
WM  1.48  10
5
R 0 m
zn cm 
i0
WM  37.5m
3 1/ 4
 42 m
is determined from a parabolic fit to the measured electron
density profile
• The plasma pressure is always much greater
than the magnetic pressure, and as such the
pinch effect can be neglected.
• Three stages of the plasma evolution have
been identified.
• There is a good agreement between the
simulated density profile and measured in
experiment.
• The results of MHD simulations allowed us to
formulate simple model.
• The matched spot size depends only on the
capillary radius, the initial ion density and the
mean ion charge
• The evolution of a hydrogen filled capillary
discharge waveguide is different compared
with previously discussed.
Capillary discharge in evacuated
channel
• Plasma inside the channel is created from the ablation
and ionization of the wall material.
• The initial conditions: after the threshold voltage is
reached, a strongly nonuniform surface breakdown
happens at the wall.
• During this stage 3D-effects and effects of plasma
quasineutrality disturbance play role.
• This stage is short in time and does not affect plasma
parameters at a later time.
• For a correct description of the discharge, it is actually
sufficient to allow only for the fact that the discharge
begins at the wall surface.
• It happens when the formula for a plasma electric
conductivity is used both for high and low temperatures.
Evacuated polyacetal capillary
Capillary radius
R0  0.6m m
The capillary is of polyacetal
Z  7, A  14, 0  1g / cm3
The capillary plasma is perturbed by a current pulse
I (t )  I 0 sin(t / t0 ), I 0  25kA, t0  220ns
Shin et al, Phys.Rev.E, 50, 1376, 1994
The electric current flows in the vicinity of the channel axis during the first
stage of the discharge, then the radial distribution of the electric current
density becomes smooth.
The radial distributions of the plasma parameters
at t = 200 ns
Typical plasma parameters distribution in the quasi-equilibrium state, when there is
both mechanical equilibrium and thermal quasi-equilibrium (Joule heating is
balanced by the heat outflow due to thermal conductivity and radiation losses).
t  t , t
t  th
Consequently, after the transition stage, the
Capillary plasma is in equilibrium.
MHD instabilities
Our model gives somewhat overestimated plasma temperature and underestimated
density of the discharge plasma
t  t h
Rem  1
In these conditions different ideal and dissipative MHD
instabilities can be expected to occur inside the channel.
These instabilities can lead to the excitation of MHD
turbulence, and a change in the transport process.
We investigated the possible influence of an anomalous transport on the
parameters of the considered capillary discharge. By incorporating the
additional turbulent transport as well as the correspondent Joule heating
we decrease the plasma temperature and increase the plasma density.
Conclusion
• The capillary discharge fills the channel
with plasma of uniform density and
slightly nonuniform temperature.
• Plasma is in quasi-equilibrium.
• The situation is unstable from the point of
view of typical MHD instabilities of Zpinches.
THE RESULTS OF MHD SIMULATIONS
Capillary diameter
0.3m m, 0.5m m, 1m m
The capillary plasma is perturbed by a current pulse
I(t)  I0 sin(t / t0 ), I0  3kA, t0  125ns
1.
2.
3.
t=5-10ns - the fast plasma compression
the ablation of the wall is due to the heat
flux from the central region
the quasi-equilibrium stage: mechanical
equilibrium (the Ampere force is balanced
by gradient of the plasma pressure) thermal
quasi-equilibrium (Joule heating is balanced
by the heat outflow due to thermal
conductivity and radiative energy losses)
Janulevich et al., J. Opt. Soc. Am. BOpt. Phys. 20, 215, 2003.
The radial distributions of electron density and temperature for different moments of time
1.
2.
3.
The density profile is concave with the minimum on the axis, which insures the
pump laser guiding.
The density on the axis increases with decrease of the capillary diameter.
For 1mm capillary
N  1018 cm3
e
For 0.5 mm capillary
Ne  2 1019 cm3
Another important application of the capillary discharges is
related to their property to form on the long time scale evolution
inside the capillary the plasma density profile with a density
minimum at the axis which provides the ultra-short laser pulse
guiding (Ehrlich, et al., 1996). In this case the capillary plays a role
an optical waveguide for a laser pulse (Tajima, 1985) providing its
propagation over a distance greater than the defocusing length.
The interaction of intense laser radiation with plasmas is important
in a wide variety of applications such as the generation of coherent
short-wavelength radiation through high-harmonic generation ,
X-ray lasers, gamma ray generation, and the development of novel
plasma-based accelerators.
For such applications the laser-plasma interaction length is
limited fundamentally by diffraction to lengths of the order of the
Rayleigh range
ZR   w20 / 
where w0
is the laser waist size.
In order to increase the distance over which the intensity of the
laser pulse is maintained at a value close to that at its focus, it is
necessary to channel the laser pulse in some manner.
In the plasma waveguides a plasma is formed with a transverse
electron density profile with a minimum on the axis of propagation,
corresponding to a transverse refractive index profile that
decreases with radius providing a focusing effect.
ne(r )  ne(0)  ne(r / R)2
A lowest order Gaussian beam will propagate through a plasma channel with a
constant spot size
1/ 4
re  e2 /mec2
WM
is the classical electron radius
 R2
 
  re ne



The laser matched waist size
WM m  1.48  10 Rm
5
1/ 2


/ ne 0  cm
3
Pondermotive and relativistic effects are neglected.
The plasma channel is not further ionized by the propagating pulse.

1/ 4
• The fast-Z-pinch (Hosokai et al.,2000), gas-filled (Spence
et al., 2001), and ablation (Ehrlich et al., 1996, Levin et
al., 2005) capillaries have been used for the laser guiding
and electron acceleration.
• The high quality GeV range energy electron beams have
been obtained in the high intense laser interaction with
plasmas generated inside the capillary filled with
hydrogen-gas (Leemans et al., 2007, Karsch et al., 2007,
Rowlandset al., 2008) and inside the ablative capillary
(Kameshima et al., 2008).
Our aim is to study the long time operation of the capillary in the
experiments on the laser electron acceleration in the plasma
formed by a discharge inside the ablative capillary.
1. The experiment set up and the results on the capillary
operation.
2. The results of magneto hydrodynamics (MHD) simulations of
the capillary discharge laser pulse guiding
3. The relativistic electron acceleration
4. Discharge effect on the capillary wall erosion.
Experiment setup
Ti:Sapphire
energy - 1 J
pulse duration 35 fs .
The focal spot size along
the x- axis
wx=32µm
the y-axis
wy= 39µm
(1)
(2)
(3)
(4)
(5)
(6)
(7)
CPA driver laser
the Nd:YAG igniter laser
the ablative capillary target
the magnetic spectrometer
the phosphor screen
The CCD camera
image intensifier, the CCD camera
The ablative capillary comprises the tube of acrylic resin of the length
equal to 4 cm with the diameter of the channel of 500µm at its axis.
12-20kV
The Nd:YAG laser pulse with the energy of
40 mJ triggers the discharge-circuit by
ionizing the inner wall material because
the acrylic resin has a zero electric
conductivity.
The abrupt resistance reduction of the
ionized inner wall material induces a
discharge inside the capillary after
approximately 100 ns the igniter YAG laser
pulse has been applied.
At this time main Ti:Sapphire laser pulse
enters the capillary
The discharge plasma life time ~ 1μs.
The ablative capillary target has apparent
advantages for the electron acceleration:
1. The fact that the ablative capillary is initially evacuated is
convenient from the point of view of the vacuum condition
keeping.
2. The use of the igniter YAG laser as discharge trigger instead of
generating a high voltage short pulse in the electric circuit
provides a stable control of the discharge initiation with an
accuracy within 10ns.
On the other hand side, the ablative capillary parameters
change during the capillary target operation. These may cause a
gradual slippage of delay time between the injection of the
igniter laser pulse and discharge start.
We assume that the conditions for the local thermodynamic equilibrium are
satisfied separately for the electron and ion components. These conditions
are used to calculate the degree of ionization.
In these simulations the capillary was taken to be made from polyacetal
with R0=250 μm.
 t 
 t 
 t3
 exp    I 2 
I (t )  I1 sin 
 2t1 
 t2 
 t1
I1  1.9kA,
I 2  0.24kA
t1  90ns,
t1  46ns,
t1  200ns,
 t

 sin 

 2t3

 t
 exp 
 t4

t1  150ns
We assumed that all plasma inside the channel was produced due to the
evaporation of capillary wall material caused by heat flux from the hot
discharge plasma to the wall. A=14, Z=7, ρ=1 g/cm3

,

Pinching capillary discharge
1. In the initial stage, fast plasma
compression, i.e., the radial pinching
of a plasma, occurs from a channel's
periphery to its axis.
2. The peak values of electron and ion
temperatures are achieved at the time
of electric current maximum. Then the
plasma temperatures decrease.
3. For t>100ns the temperatures change
smoothly with time. The radial
distributions of plasma density and
temperature inside the channel
become smooth.
4. The electric current flows in the
vicinity of the channel axis during the
first stage of the discharge, then the
radial distribution of the electron
current density becomes smooth.
Temporal evolution of the electron density and
temperature on the axis
The radial distributions of electron temperature and density at moments
t=100ns and t=300ns
1. The plasma pressure is almost constant across the capillary crosssection after the short (~60 ns) initial stage and is much higher than
the pressure of the magnetic field generated by the main discharge
current.
2. Thus the Ampere force can be neglected and the capillary plasma is
confined radially mainly due to the capillary walls.
3. The maximum of Te at the axis leads to a minimum ne. Electron
temperature Te decreases due to heat transport to the capillary wall
during all the current pulse except the short initial stage.
Conclusions
1. Dense (ne ~ 1018cm-3) stable plasmas with
temperatures of several eV can be generated in
evaporating-wall capillary discharge.
2. This plasma column exhibits spatial reproducibility
from shot to shot.
3. Such discharges provide a convenient source of
dense highly ionized plasma with concave radial
profile, which allows to create better conditions for the
optical guiding of ultra high laser intensities.
Optical Guiding
of laser pulse is controlled by adjusting the time of the plasma formation and
of the main pulse arrival at the target.
ne ~ 1018cm-3
Images of transmitted laser pulses at the exit of the capillary target taken by the CCD
when the time matching condition between the injection and main laser pulses is fulfilled.
1. At a few μJ input energy level the spot size of the guided laser is approximately
equal to the initial focal spot size at the capillary entrance.
2.The transmitted laser pulse for the 1~J input energy level has two times wider spot at
the exit with the same pattern as observed in the previous case with lower laser energy.
When the time matching condition is not respected, the transmitted
light does not show a good collimation, and its intensity is of
one-seventh compared the good guiding case.
Multi MeV quasi-mono energetic electron
bunch generation
During the laser pulse interaction with the capillary plasma the
relativistic electrons are accelerated and the electron beam
accompanies the transmitted light.
The electron beam is deflected by the
magnetic spectrometer (4 ), hits the
phosphor screen (5) generating the
scintillation light and producing the image
on the screen.
The pattern of the image on the phosphor
screen in horizontal and vertical directions
with respect to the incidence provides the
information on the electron energy spectrum
and on their angular divergence.
The electron distribution in the energy-angle plane
ne ~ 2.2×1018cm-3
The central energy of the electron
bunch is equal to 18~MeV with
±11% energy spread
The divergence angle~ 12.5 mrad
The laser was injected at the 125 ns delayed from discharge start and a 1~J energy
injection. The phosphor screen image with energy-divergent angle scales(a) ; The
energy spectrum in the integral of the divergent angle over the 1\e2 intensity(b); The
divergent angle distribution in the integral of the energy over 1\e2 intensity(c).
ne ~ 9.0×1018cm-3
Broad spectrum electrons with
wider angle distribution
The energy spectrum can be approximated by the maxwellian distribution
with effective temperature about ~100 MeV. The maximal electron
energy is equal to ~ 300 MeV.
Counterplay between the laser pulse guiding
and the electron generation
• In the case when the electron signal is distinctly seen on the
phosphor screen, the transmitted laser light is not detected. In total
contrast to that when we see a high intensity light transmission
through the capillary the electron signal disappears.
• This behavior is related to the process of the electron injection to
the acceleration phase in the wake wave. A good laser guiding
happens when the plasma density inside the capillary is relatively
low, which in its turn corresponds to the unfavorable conditions for
the wake wave breaking with no injection .
• In the opposite case of relatively high plasma density we have the
wake wave breaking, which causes the electron injection and their
acceleration. The laser depletion length becomes shorter than the
capillary length and the laser pulse does not reach the capillary exit
.
Wall conditioning and erosion
Repeated discharges and multiple events of high
intense laser-capillary interaction gradually change
the capillary parameters. We expect that as a result of
a repeated ablation
• the capillary diameter increases
• appears heterogeneity and irregularity in the capillary
wall surface.
This can
1. change the time between the igniter pulse injection
and the discharge development
2. modify the plasma distribution and to affect the laser
pulse propagation.
d~500µm
(a) - an unused capillary
(b) - the capillary has experienced 39 discharges and interactions with
the 1J laser
(c) - capillary had 140 discharges accompanied by the 1 J laser pulse
• The experiments on the laser electron acceleration in
•
•
•
•
the ablative capillary plasma have been carried out.
As a result of the laser ignited discharge
development the plasma channel is formed by the
discharge inside initially evacuated capillary.
We demonstrate the high intense short laser pulse
guiding over the 4 cm length with a constant focus
spot size.
The generated relativistic electrons show both the
quasi-monoenergetic and quasi-maxwellian energy
spectra.
The analysis of the inner walls of the capillaries that
have operated for several ten shots show the wall
deformation and filling with the blisters resulted from
the discharge and laser pulse effects.