Simulations of capillary discharges

Prague, November, 2012

• 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.

INTRODUCTION

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 coherentshort-wavelength radiation through high-harmonic generation ,X-ray lasers, gamma ray generation, and the development of novelplasma-based accelerators.

For such applications the laser-plasma interaction length islimited fundamentally by diffraction to lengths of the order of the Rayleigh rangewhere 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.

/wZ 20R

0w

Table-top soft x-ray lasers 

J. J. Rocca, Rev. Sci. Instr. 70, 3799 (1999)

On axis spectra of Cd capillary discharge plasma emission from 13.2

nm line of Ni-like Cd

Pin-hole images of the soft-x-ray emitting region

Schematic view of setupElectric current pulse

Simulation

• A number of important applications, such as short-wavelength lasers and novel schemes for particle acceleration, involve the interaction with plasmas of ultra-short laser pulses with peak intensities in the range

• 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.

16 24 210 /W cm

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 length1 GeV beam: a0 ~ 1.46 (40 TW, 37 fs)

Peak energy: 1000 MeVDivergence(rms): 2.0 mradEnergy spread (rms): 2.5%Charge: > 30.0 pC

W.P.Leemans et al, Nature Physics, 418 (2006)

CAPILLARY

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 initially-evacuated 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 modelWe use the approximation of two-temperature, one-fluid MHD.

Owing to the large length-to-radius ratio of the capillary

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 theplasma (pinch effect).

l R/ 0 1

Important dissipative processes: electron and ion thermal conductivities, Joule heating, Nernst and Ettinghausen effects, 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

0,0,

0, ,0

,0,0

z

z

j j

B B

v v

( ) ( )

2

,

1( )

4

z

zz

e

e ee

z ze e

e

B Ec

t rvBR

Een c

vBTE j B

en en r c

B TcvB rB c B

t r r r r r r

N

System of equationsSystem of equations

10

1 1

1

1 1

rr rr

e e ee r ei i e

i i ii ei e i rr

r vt r r

v v pv jB

t r r c r r

B B cv E

t r r r r r

pv r v jE r q Q C T T

t r r r r r

pv r v jE r q C T T

t r r r r r

rr

vr v

r r r

Dissipative coefficientsDissipative coefficients

20 0 2

2 22 ,

3 3

, ,

rr

e ee e e

ii i

v v vr

r r r

T TjE B q BT j

r r

Tq

r

N N

1 4

1 15 02

1, , , ,

1 , , 3 , 0.96

e ee e e

e e e ei

e ei ee ei e ei i i ii

e A

n Tx w x w

m m c

m mx w C n n T

e n Am

N

,ee ei

eBx

m c

22ee

ei

wz

.2,1;7,,2,1,

,6,,2,1,,

0,

1,

22,

33,,

22,7

21,7

4

2,2

1,

jiwwww

iwxwx

wxwwx

jijijijiji

ee

ieiei

Expressions for lji, are obtained in the so called two-polynominal approximation

wxei , are determined with the accuracy of a few percent, which is quite sufficientbecause the accuracy of the Landau approximation for the collision integralis of the order of ./1 ei

The values of 7,,1, iwxei are calculated by Braginskii for for certain

multiples of 12 w which is not very convenient when w runs through a continuous set of values. For example, values of

2, ji and 2/1, ji differ by more than a factor of 2.

We take into account the possible considerable difference between ee and ekotherwise the accuracy of the resulting system of equations is incorrectly reduced.

Neutral particlesNeutral particles

3 32 2

4 4 42 1

1 1

4 2 4, ,

3 3

для 1, 1 для 1

e ei i iiei ii

e e A ei

e n z e n z

m T Am T

z z z z z

2

2

4

42

0

1

1

1 4,

34 1 /

ei ei en

ei ei

een o

e

z для z

z z для z

m ez

z T

Equation of state and degree of ionizationEquation of state and degree of ionization

6z For the equation of state and the ionization degree, the approximation of local thermodynamic equilibrium is used separately for the electron and ion components.

0z z

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 shellZ/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

0

60

1

10

z z

z

the simplified Saha formula is used, taking only neutral and once ionized atoms .

Thermodynamic properties Thermodynamic properties

3/ 2

2

2, 1 ln

2e e e

e eA A

U zzT m T VF V T

Am z Am

V – удельный объем одного иона

,

,

, , 0

1, , ,

,

3,

2

e

e

ee e

V T

ee e e e e e

A e V

ee e

T

ii i i

FV T z V T

z

FT F V T T V T

Am T

Fp V T

V

TT p

V

Boundary conditionsBoundary conditions

exr R Free boundary

0 0

,

0 ,4

0

2,

ex

ex

exex

e i ex

R

e ee e ex ex

R

i

exex

dR tv R t t

dt

v vp p p t

r r

T c BT rB или T R t t T t

r r r

T

r

I tB R t t

cR t

N

0r 0 0, 0 0, 0 0 0 , 0e ie

T Tv r B r q r

t t

1. Радиуса капилляра2. Параметров внешней электрической цепи3. Материала стенок4. 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 пренебречь2. Испарение стенок капилляра

Различные типы динамики капиллярного разрядаРазличные типы динамики капиллярного разряда(физические процессы, которые определяют тип (физические процессы, которые определяют тип

динамики плазмы)динамики плазмы)

Динамика плазмы в капиллярных разрядах зависит от нескольких параметров:

Пинчевой капиллярный разрядПинчевой капиллярный разряд

0 2R мм

6 30 1.37 10 /г cм

307, 14, 1 /Z A г cм

Радиус

Начальная плотность Ar

Капилляр из полиацетата

0 0 0 0( ) sin( / ), 40 , 60I t I t t I kA t нс

Электрический ток

Столкновительное возбуждение неоноподобных ионов аргона

19 30.3 1.0 10 , 60 80e en см T эВ

Рокка и др. 1994

0 10 20 30 40 50 600

1

2

rad

ius,

mm

0 10 20 30 40 50 600

1

2

rad

ius,

mm

0 10 20 30 40 50 60

tim e, ns

0

1

2

rad

ius,

mm

1 0

1 7 . 1

1 7 . 9

1 8 . 8

1 9 . 3

1 9 . 7

0

13

26

39

52

65

0

10

20

30

40

Lo

g10

of

elec

tr.

den

sity

in c

m-3

Te

, eV

elec

tric

cu

rren

tkA

Моделирование пинчевого капиллярного разрядаМоделирование пинчевого капиллярного разряда

0

1E+19

2E+19

3E+19

4E+19

5E+19

Ne

, cm

-3

0

20

40

60

80

Te

, eV

0 10 20 30 40 50 60tim e, ns

1/ 2

0 0/ , / 4c A At R v v B

t> 20нс 30-40% I около стенки<50% I по аргоновой плазме5% 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

jpTe ,,,1.The increase of on the axis.

2. I is separated in two spatiallydistinct 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

19 34 10en cm

60eT 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 . 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.

1016 W / cm2

Hydrogen capillary (nonpinching discharge)

R m I A t ns p mbar0 0 0 0150 250 200 67 , , ,

Alumina capillary filled with hydrogen

Three stages of the plasma evolution

1. Magnetic field penetrates the plasma ona time scale of 1ns. Pinching of the plasma isnegligibly weak. The plasma is heated and ionized locally. Radial distributions of plasma parameters are homogeneous. 50ns2. A redistribution of the plasma temperatureand density occurs during t=50-80ns. Thermalconduction becomes significant.3. Quasi-state equilibrium at a given electric current

p H 2 8/

t ns t ns t nsh 5 10 2, ,

The plasma temperature has its maximum on the axis, because the Ohmicheating is balanced mainly by thermal conduction to cold wall. This results inan axial minimum in the electron density profile.

Temporal evolution of the axial electron temperature and density

0 50 100 150t, ns

0

1

2

3

n e , 1

018 c

m-3

0

2

4

6

8

10

Te ,

eV

ne

Te

After t=80ns the axial electrondensity is constant. The axialelectron temperature slowlydecreases with time.

At t=60ns

n Te e 2 8 10 3 418. , .cm eV-3

Experimental value ne 2 7 1018. cm-3

The measured and calculated electron density profiles at t=60ns

-150 -100 -50 0 50 100 150d ,m

1.5

2

2.5

3

3.5

4

4.5

n e, 10

18 cm

-3

Measured profile corresponds to an average electron density profile over several ns.Allowing for errors in measurement of the time t of approximately ns, and averaging over the duration of the probe pulse used in the measurements, the simulations are in good agreement with the measurements.

t=55 nst=60 ns

t=65 ns

5

Direct heating of the lattice due to ion-lattice thermal conduction plays

only a secondary role.

The thermal flux due toelectron thermal conduction

Partial ionization of the wall andheating of the free electrons

The lattice heating by energy exchange with free electrons

For t=100-150ns, the degree of ionization of atoms in the wall materialin a layer depth.

10 4

08. m

Te 18. eV Lattice temperature =

For practical applications of H-filled capillary discharge waveguides, it is important that the capillary has a long lifetime.

30

C at t = 50ns

200 C at t = 100ns

500 C at t = 150ns

no melting ordisruption

In experiment after discharge pulses the increase in capillary diameter was105 1m.

Simple model of plasma equilibrium

• 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

• There is no difference between the electron and ion temperatures

For t>80 ns the following conditions are fulfilled:

xe Be ei / .0 04

iT

The equation for heat flow 0Eσdr

dTrκ

dr

d

r

1 2e

dT

drT T

rr

0

00, boundary conditions

eT

Assuming Coulomb logarithms to be constant,

e T 05 2/

03 2T /

electric conductivity

electron thermal conductivity

Assuming that e r e r RT T

0 0

we consider T 0

Introducing a new variable r R/ 0and new function

u( ) determined by

7/22/1

022

0 u2/ER 07)T(

we obtain

0u,0ddu

,uddu

dd1

10

7/3

The unique nontrivial solution of this equation is found numerically.

Assuming Coulomb logarithms to be constant,

e T 05 2/

03 2T /

electric conductivity

electron thermal conductivity

Assuming that e r e r RT T

0 0

we consider T 0

Introducing a new variable r R/ 0and new function

u( ) determined by

7/22/1

022

0 u2/ER 07)T(

we obtain

0u,0ddu

,uddu

dd1

10

7/3

The unique nontrivial solution of this equation is found numerically.

Assuming Coulomb logarithms to be constant,

e T 05 2/

03 2T /

electric conductivity

electron thermal conductivity

Assuming that e r e r RT T

0 0

we consider T 0

Introducing a new variable r R/ 0and new function

u( ) determined by

7/22/1

022

0 u2/ER 07)T(

we obtain

0u,0ddu

,uddu

dd1

10

7/3

The unique nontrivial solution of this equation is found numerically.

0 0.2 0.4 0.6 0.8 1

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

u

107.01'u,067.0)0(u

The pressure is constant across the capillary 0T02nrTr2np ee

The electron density profile

2

0

2

e

7/2

ee Rr

33.010nru0u

0nrn

The axial electron density

7364.0

0um21

n0n

7/20e

e

55.1dum1

0

7/20

0 50 100 150 200r, m

0

1

2

3

4

5

6

7

8T

e , eV

0

1

2

3

4

5

6

n e ,

1018

cm

-3

Te

ne

simulation

simple model

t=100ns

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 with a constant spot size2/r)0(n)0(n)r(n 2''

eee

4/1

"ee

M )0(nr2

W

- the classical electron radius

From the equilibrium model

m42cmzn

mR1048.1W 4/13

0i

05M

is determined from a parabolic fit to the measured electron density profile

er

m5.37WM

• 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

mmR 6.00

30 /1,14,7 cmgAZ

Capillary radius

The capillary is of polyacetal

nstkAIttItI 220,25),/sin()( 0000

The capillary plasma is perturbed by a current pulse

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

ttt ,

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).

htt

Consequently, after the transition stage, theCapillary plasma is in equilibrium.

MHD instabilities

htt

Our model gives somewhat overestimated plasma temperature and underestimateddensity of the discharge plasma

1Re m

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 MHDturbulence, 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 heatingwe 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 Z-pinches.

THE RESULTS OF MHD SIMULATIONS

mmmmmm 1,5.0,3.0Capillary diameter

ns125t,kA3I),t/tsin(I)t(I 0000 The capillary plasma is perturbed by a current pulse

1. t=5-10ns - the fast plasma compression2. the ablation of the wall is due to the heat flux from the central region3. 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. B-Opt. Phys. 20, 215, 2003.

The radial distributions of electron density and temperature for different moments of time

1. The density profile is concave with the minimum on the axis, which insures the pump laser guiding.2. The density on the axis increases with decrease of the capillary diameter.3. For 1mm capillary

For 0.5 mm capillary

31810 cmNe

319102 cmNe

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 coherentshort-wavelength radiation through high-harmonic generation ,X-ray lasers, gamma ray generation, and the development of novelplasma-based accelerators.

For such applications the laser-plasma interaction length islimited fundamentally by diffraction to lengths of the order of the Rayleigh rangewhere 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.

/wZ 20R

0w

2)/()0()( Rrnnrn eee

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.

A lowest order Gaussian beam will propagate through a plasma channel with a constant spot size 4/12

ee

M nr

RW

2e

2e cm/er is the classical electron radius

The laser matched waist size

4/132/15 0/1048.1 cmnmRmW eM

Pondermotive and relativistic effects are neglected.The plasma channel is not further ionized by the propagating pulse.

• 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).

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.

.

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.

Experiment setup

(1) CPA driver laser (2) the Nd:YAG igniter laser (3) the ablative capillary target (4) the magnetic spectrometer (5) the phosphor screen (6) The CCD camera(7) image intensifier, the CCD camera

Ti:Sapphire energy - 1 J pulse duration 35 fs .

The focal spot size along the x- axis wx=32µm the y-axis wy= 39µm

The ablative capillary comprises the tube of acrylic resin of the lengthequal 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 aresatisfied separately for the electron and ion components. These conditionsare used to calculate the degree of ionization.

In these simulations the capillary was taken to be made from polyacetal with R0=250 μm.

nstnstnstnst

kAIkAI

t

t

t

t

t

tI

t

t

t

tItI

150,200,46,90

24.0,9.1

,exp2

sinexp2

sin)(

1111

21

431

32

211

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

1. The plasma pressure is almost constant across the capillary cross-section 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.

The radial distributions of electron temperature and density at moments t=100ns and t=300ns

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 Guidingof 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 approximatelyequal 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 phosphorscreen 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.

(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

d~500µm

• 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.