Laser plasma density profile modification by ponderomotive force

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Laser plasma density profile modification by ponderomotive force

R. Dragila, J. Krepelka

To cite this version:

R. Dragila, J. Krepelka. Laser plasma density profile modification by ponderomotive force. Journal

de Physique, 1978, 39 (6), pp.617-623. �10.1051/jphys:01978003906061700�. �jpa-00208793�

LASER PLASMA DENSITY PROFILE MODIFICATION

BY PONDEROMOTIVE FORCE (*)

R. DRAGILA and J. KREPELKA

Faculty

of Nuclear Science and

Physical Engineering,

Department

^of

Physical

Electronics, ^Brehova 7, ^{115 19 Praha}

1, Tchécoslovaquie (Reçu

^{le 23 décembre}

1977, accepté

^le

22 février 1978)

Résumé. ²⁰¹⁴ La modification du profil de la densité du plasma

produite

par la force pondéro-

motrice est analysée par ^un traitement numérique ^du système auto-cohérent des équations hydro-

dynamiques

^{et des} équations de Maxwell dans une géométrie plane. ^On prévoit des oscillations du

pouvoir

réflecteur en fonction du temps ^{à cause} des oscillations du profil ^{de densité au cutoff.}

Abstract. ²⁰¹⁴ The plasma

density profile

modification due to the

ponderomotive

^{force is}

analysed

by ^{numerical treatment of a} consistent system

of hydrodynamical

^{and Maxwell equations in a} plane geometry. Reflectance temporal oscillations ^are proposed ^{as a result of} density profile oscillations at the cutoff.

Classification Physics ^Abstracts 52.50J ^- 52.40D - 52.65

1. Introduction. ^- In

present

laser fusion _expe- riments the conditions under which the

pondero-

motive force

[1]

^{dominates the} thermokinetic _pressure

are obtained. In such a case, noticeable

changes

ⁱⁿ

plasma density profile

^and

expansion dynamics

^{can be}

expected [2]. Experimentally, plasma density

^variations

have been observed ^at the cut-off

region by

^Zakha-

renkov et al.

[3]

^and

profile steepening indirectly

evidenced

by

Manes et al.

[4]

from observations the

degree

^of

polarization

rotation of the backscattered

light.

The noticeable

steepening

^{of the}

density profile

can

produce

substantial modifications in

absorption

processes and can

exchange

the role of

parametric

and resonance processes ^at

oblique

^incidence

[5].

Using

^{the WKB}

approximation

ⁱⁿ the under-dense

region

^and

estimating

the electric field

intensity

^{at the}

cut-off

neighbourhood,

where the WKB

approxi-

mation

fails,

Brueckner and Janda

[6]

simulated the

dynamics

of irradiated

pellets

with results

qualitatively

close to those of Lee et al.

[7],

^{where a}

locally stationary density profile

^{in the frame}

moving

^{with the}

density step assuming

isothermal flow was outlined

analy- tically

^{for a}

plane geometry.

Mulser and van Kessel

[8]

analytically analysed

^a

steady-state spherical

^model

solving plasma continuity

^{and motion}

equations together

^{with a wave}

equation omitting absorption,

(*) ^This paper has been reported ^{at llth} European ^Conference

on Laser Interaction with Matter, ^Oxford 19/23 September ^1977.

however Virmont et al.

[9]

^have shown that these

profiles

^{are unstable} except ^for very small laser fields.

The correct treatment of the

phenomena

^considered

requires

the consistent system ^of

gasdynamical

^and

Maxwell

equations

^to be solved

simultaneously, including

^the

ponderomotive

^{force in the momentum}

and _energy balance

equations

^for system

plasma-

radiation.

2.

Physical

^{model. - For laser radiation fluxes}

in a range qo z

1013-1016

W

cm-2

the mass

velocity

of the

expanding

^{corona is of order of}

magnitude

of u N 107-108 cm s -1

and, consequently,

^{the zero}

order

approximation

^{in a small} parameter

ule «

¹

(c-velocity

^of

light)

^used

throughout

^the present paper is

justified

^and

yields

^the

ponderomotive

^{force in the}

form

(Dragila 1978)

e-dielectric constant, E-electric field

intensity.

^Such

the force will dominate a thermokinetic _pressure, if

where _p ⁼ _{n. mi} is mass

density,

m; ion mass, T tem-

perature, pe critical

density

^{and where} pressure has been assumed to scale with A

spatially,

^{at least at some}

cutoff

neighbourhood.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003906061700

618

Assuming

^{a weak}

absorption

^{in the corona, which}

is consistent with the

expected

steep

density

pro- file and normal incidence of

radiation,

^i.e.

setting e 1/2E 2 E5,

the condition for the threshold radia- tion flux

density

qo,, for the

ponderomotive

^{force to}

dominate at the cutoff

neighbourhood,

^reads

where > Â

indicates a

spatial

average ^over

Â, providing

^the

density gradient

characteristic

length

is of the same order of

magnitude. Thus,

^{for Nd}

glass

laser

(n,, * ¹⁰²¹ cm -3, T, ;:t 1

^{keV for a} range of laser fluxes

considered, 1 e 1 >’/’ ;:t 0.1)

^{it becomes}

while for

C02

^laser

To

verify

the role of the

ponderomotive

^{force in the}

dynamics

^{of a} laser-induced

plasma,

^the

fully

^ionized

planar plasma layer

with initial thickness z ⁼ zo and

homogeneous density p(z, t

⁼

0)

⁼ po, 0 z , zo is assumed to be irradiated

by

^laser

light represented by

^an

electromagnetic plane

^wave

where

X(z)

represents the electric or

magnetic

^field

component ^{and co is an}

angular frequency.

^The

dynamics

^of the irradiated

plasma

has been described

by

^{means of a one-fluid}

two-temperature Lagrangian

model

including

^electron pe and

ion p;

pressures, real and artificial

viscosities,

^{electron heat conducti-}

vity,

electron-ion

relaxation,

^laser

light

energy

absorp-

tion as the time

averaged

^over

2 n/co

^{Joule heat source}

term 2 0’ 1 E l’

^{in the electron} energy balance _equa- tion and time

averaged

^{over 2}

nlw ponderomotive

force

(1)

^{in the momentum} conservation law

(3).

Consequently,

^the

hydrodynamic equations

^{are as}

follows :

(in

^CGS

units, Ta

ⁱⁿ

keV),

^{u is the}

velocity, e,,,

^the spe- cific internal _energy

(related

^{to unit}

mass), Ta

tempe- rature, ^{E the} electric field

intensity, Ba

^the

specific

^heat

capacity,

xe, 110’

Qo

coefficients of thermal conduc-

tivity,

^real

viscosity

^and electron-ion

relaxation, respectively,

y

= -i (Poisson’s constant),

^{A is the}

atomic

weight,

^{Z the}

degree

^of

ionization,

^{ln A the} Coulomb

logarithm, Qe

^and

Qi

^are artificial electron and ion

quadratic

viscosities

[10],

^{VI = 1 so that there}

is no artificial

viscosity

^{included at} the rarefaction wave, and Am is the mass of the cell in the

Lagrangian

différence scheme.

For a

plane

geometry,

T,, > Ti typically, hence, according

^{to the relation}

xo;/xoe

⁼

(me/mJI/2,

^{the ion}

heat conduction can

usually

^be

neglected (me, mi

^are

the electron and ion

masses).

According

^{to the}

relation th > tm 2 nlOJ

^between

the

hydrodynamical th

^and

Maxwellian tM

^time

scaling,

^Maxwell

equations

^{as well as}

electromagnetic

energy conservation law can be treated in the

quasi-

stationary

^{form as}

with E ⁼

Ey

^{and H}

^{= Hx}

^the components of electric E ⁼

(0, Ey, ⁰⁾

^and

^magnetic

^{H =}

^{(Hx, 0, 0) fields,}

e ⁼ el ⁺

’82 complex

dielectric

function,

el = 1 -

pl Pc,

’62

p V e/w3, 0’

⁼

we2/4

^rc

conductivity,

p,,, critical

density,

cop, ^{and ve electron}

plasma

^{and collision}

frequencies, respectively.

Introduced in

(6)

^{the effective}

conductivity

represents ^the energy

exchange

between the

plasma

and

electromagnetic

field due to the _power of a

ponderomotive

^force

[11].

^The

equations (5)

^{have been}

solved

using

^the method evaluated

by Afanasyev

et al.

[12],

^i.e.

introducing

the reflectance

V(z) by

^the

relation

where

X+, X-

^are incident and reflected waves res-

pectively.

^{The collision}

frequency

ve should be understood in a

generalized

^{sense, i.e. as}

where

e and _me are the electron

charge

^and mass,

and

Here V, (i)

represents ^the effective contribution to the

absorption

processes of the

possible

instabilities

arising

^at the cutoff

neighbourhood (

pi, p,

>, if

^the

relation _qo > qth for the laser ^flux

density

qo

holds,

with _qth the threshold for the

instability

under consi- deration. In such a way, ^our model treats both collision

(ve

⁼

V("» absorption by

^inverse

bremsstrahlung

^and

anomalous

heating (ve

⁼

v(i».

^A

complete

^{table of}

Vef, qth, Pi and _P2 can be found in

[12]. However,

^the numerical calculations have shown that for

parametric decay instability

^was dominant with

However,

^{in an}

inhomogeneous plasma,

the thresh- old for the

instability

considered becomes an increas-

ing

^{function of a}

density gradient.

^At the cutoff

neighbourhood

^{it is}

typically T,, > T;,

^{and thus the}

increasing

of the threshold is

effectively

introduced

by reducing

the anomalous

absorption region (

pl,

P2 >

geometrical

width where the

frequency

^{and wave-}

vector

matching

conditions for

parametric instability

to arise is fulfilled

[ 13].

The

bremsstrahlung

radiation losses can be

neglected

because the absorbed _{energy in} unit volume and unit time dominate the power of the radiation losses from the unit volume

[14],

^if

Hence,

^for

Te

^{= 10 keV and hm = 1 eV}

(Nd laser)

it _{appears q.,} ⁼ 1012 W

cm - 2,

while it is

typically T,,

1-10 keV in

experiments,

^where

3.

Computational

^{scheme. - The} consistent system of

gasdynamic (2-5)

and Maxwell

(6) equations

^has

been

numerically

^solved

using

the finite difference

implicit fully

conservative

scheme,

which is the modification of code used

by Afanasyev

^{et al.}

[12].

Computing

the field

spatial profile,

^the

absorbing plasma

^{has been}

split

^{into two}

regions :

a) (zo,

Zi 1

zc)

where the WKB

approximation

holds and

b) (zl,

Z2 >

zc)

^{the cutoff}

neighbourhood,

^where

the WKB

approximation

^fails.

(zo, Zc correspond

^{to the}

plasma

^vacuum

boundary

and the cutoff

plane, respectively),

^{to save} compu- tation time. The treatment of Maxwell

equations (6)

in the

region (

z l’

Z 2 >

^{needs a fine}

spatial

^mesh

with a

step Azm « Â,

while in the

region zo, z 1 >

where the WKB

approximation

^{is valid a}

spatial step AZWKB Â

is sufficient.

As the scale

length

^{of the}

ponderomotive

^force

(1) spatial

variation is of order of the local radiation

wavelength Â,

^the

gasdynamic spatial step

^{should be}

AzG

^{Â in the}

region (zo, z2),

ⁱⁿ

general,

^{or at least}

in the

region

^{where the}

ponderomotive

force domi- nates the thermokinetic pressure. For

instance,

^such

a

region

^can be identified with a Maxwellian

region (zi, z2)

^with

typically (z2 - ZI) ’;:t (3 : 6)

^,1. In par- ticular, each

gasdynamic Lagrangian

^cell

(initial

^size

of which was of order  and

density po) in

^the

region (z l’ Z2)

^is

successively split

^{so that the}

resulting

^cells

fulfil the local condition

AzG

^{Â and}

integrated

^back

to one

big

^{cell after}

passing

^{the discussed zone,}

taking

care about mass, momentum and _energy conservation.

4, Results and conclusions. ^- In all the numerical simulation further discussed, the

planar CD2

target with initial

density

po ⁼

1.0 g cm - j

^{and laser}

light

with a

wavelength Âo

^{= 1.06} pm has been considered.

The

trapezium

form of the laser

pulse

has been chosen with a standard rise-time _LR ⁼ _{33 ps.}

One of the

quantities

^{that can be}

experimentally

verified which indicates the processes that

directly

^or

indirectly modify

^the

plasma

parameters ^{at the cutoff}

620

neighbourhood

^{is a} coefficient R of

reflexivity,

^the

time-dependence

of which is

plotted

^on

figure

^1.

A noticeable difference can be observed between dashed and full curves

corresponding

^{to the same}

FIG. 1. ^- Reflectance versus time, omitting ( ) and including (- - - -, ....) ponderomotive ^force (PMF). (-..-) correspond ^to

the case of only classical inverse bremsstrahlung (CIB) absorption

mechanism.

radiation flux

density

qo ⁼

1014 W cm-2

but without and

including (PMF)

^the

ponderomotive force,

^res-

pectively.

The relevant

portion

of absorbed

energies

were

and

Besides the increase in reflected _energy,

switching

on the

ponderomotive

^force

brings

about the charac- teristic

temporal

oscillations of the coefficient R of

reflectivity

^{due to the}

density profile

oscillations at the cutoff

neighbourhood,

^{as will be shown in what}

follows.

Consequently,

^the

period

^of these oscillations

can be

expected

^{to be} of the order of

magnitude

^of

where the local

wavelength Â

represents ^the

density gradient

characteristic

length

^and aPMF

(8)

^{an addi-}

tional

plasma

acceleration due to the

ponderomotive

force. _{For qo} ⁼

1 O 14 W cm - 2, tos N 60 ps

^{in a}

rough

agreement with the simulation results on

figure

^{1. In a real}

experiment,

^the

expanding plasma

flow is not

planar

ⁱⁿ

general,

^{hence the}

velocity

Uc is

supposed

^to decrease and tose ^to increase in

comparison

with the

purely planar

^{case with the same laser flux}

density

^on the critical surface. The

non-regular

variations in oscillation

amplitude

^{contain no}

physics

and _appear as a result

of splining.

The substantional fluctuations in

reflectivity

^have

also been

predicted by

Brueckner and Janda

[6], however,

^{as a} result of fluctuations in the critical

density

^radius

(spherical model). Experimentally,

^the

temporal

oscillations in

reflectivity

^{have been observed}

by

Basov et al.

[15],

^{with a} modulation of order of

30-50 %

^{of the maximum value and}

period

tose = ^0.6 ± 0.1 ^ns for Al

planar target

^{and Nd}

glass

laser flux

density

qo = 7 x

1012

W cm-2 on the target ^and

explained

^{as due to}

temporal

oscillations

of parametric instability

^{induced non}

steady

^turbulent

noise.

The second effect

investigated

^{has been the}

density profile

variations due to thé

ponderomotive

^force.

For

illustration,

^{two instants}

of time tl

^{1 =} 112 ps and t2 ^{= 152} ps

corresponding

^to maximum and minimum of R for _qo ⁼

1014Wcm-2,

have been chosen and relevant

density profiles plotted

^on

figures

^{2 and 3.}

In agreement ^{with our}

previous

estimation

(see

section

2),

the threshold radiation flux

density

^{for the}

ponderomotive

^{force to} dominate is somewhere between

1014-1015

W cm-2

(Fig. 2a, 3a).

^{The relevant}

details of the

density profiles

^{within a few}

wavelengths

of the cutoff

neighbourhood

^{shown in}

figures 2b,

^3b

offer an

interpretation

^{of the}

reflectivity oscillations, namely,

^the

matching

^conditions

for the

parametric decay instability

^{are fulfilled}

in a rather wider cutoff

neighbourhood

^{Az at the}

FIG. 2. - Plasma spatial density profile ^{at t =} 112. ps omitting ( and including (- - - -, ....) ponderomotive ^force (PMF).

a) Complete profiles ; b) Detail of a relevant few wavelengths

cutoff (z ⁼ zj neighbourhood.

FIG. 3. ^- Plasma spatial density profile ^{at t = 152} ps omitting (-) ^and including (- - - -, ....) ponderomotive ^force (PMF).

a) Complete profiles ; b) Detail of a relevant few wavelengths cutoff (z ⁼ Zc neighbourhood).

instant

t2(Oz

⁼

Oz2)

^{than at} the instant

ti (Az Az,).

For

instance,

^the

matching

^condition

An,, ,ln,, 0. 1

holds for

Azi = /)o

^but

ez2 x 10 Ào. However,

^if

only

collisional inverse

bremsstrahlung absorption

^is

included

(CIB),

the reflectance oscillations are also observed

(Fig. 1)

^with

higher time-averaged

^over tosse

magnitude of ; R ), particularly,

In this _case, due to a lower

temperature Te

^{at the}

cutoff,

the

density profile

^{must be} steeper for the thermo- kinetic _pressure to compensate ^the

ponderomotive force,

ⁱⁿ

comparison

^{with the case} where anomalous

absorption

^{was taken into account.}

Oscillating

^{such the}

steepening modify

^even collisional

absorption

^effi-

ciency [4]

^to oscillate. The oscillations

alone,

^{for both}

classical and anomalous

absorption mechanisms,

appears ^to be a result of temperature

temporal

oscillation

(Fig. 4),

^{and thus}

competition

between the

ponderomotive

force and the thermokinetic _pressure.

Once the

ponderomotive

^force

steepens

^the

density profile,

^the

integral

coefficient

of reflectivity

^decreases,

Fm. 4. ^- Electron T_ ^{and ion} Tic temperature ^at the cutoff versus

time.

however,

the local

absorption

coefficient does not

change

^at the cutoff and thus less

particles

^{are heated}

at the same rate, ^{i.e. to}

higher temperature.

^Conse-

quently

the thermokinetic _pressure increases which leads to the

plasma expansion

^and

cooling

^{at the}

cutoff, decreasing integral

coefficient of

reflectivity, dominating ponderomotive force,

^etc.

Also,

^the

velocity profile

^{has been}

investigated

^to

verify

^the relation of the

ponderomotive

^{force to}

underdense

plasma

acceleration. As can be seen from

figure 5,

^{a small amount of rare}

(p « pj plasma

^is

blown off from the cutoff when

qo >

qc simulta-

neously during

^the

forming

^{of the}

steepened density

FIG. 5. ^- Velocity spatial profiles ^at cutoff (z ⁼ zc) neighbourhood omitting (-) and including (- - - -, .... ) ponderomotive ^force

(PMF).

profile.

^{If the}

ponderomotive

^{force is not considered}

and

formally

^{the same}

absorption

^rate assumed for qo ⁼

101-5

W

cm - 2,

^the

velocity profile

^{would corres-}

pond

^{to the}

prolonged straight

part ^{of the dotted}

curve

(Fig. 5).

^{In such a} way, ^our calculations are

complementary

^{to those of}

Campbell

^{et al.}

[16]

^where

the electrostatic acceleration of ions was

investigated

(KMS

Fusion code

TRHYD).

622

Most of the calculations

presented

^{here have also}

been done for

CO2

^laser

(îo

^{= 10.6}

ym)

^irradiated

targets

^to

verify

^the

sensitivity

of the studied

phe-

nomena to the

wavelength.

^{We will not} present ^these

parallel

^results

here,

^{as these can be}

simply

^{related to}

the results

corresponding

^to

Â.

⁼ 1.06 gm.

According

to

(1),

^the

plasma acceleration, assuming

^{a weak}

absorption,

which is consistent with the condition qo > q,,,

(see figure 1),

^becomes

where

Eo

^{is an} electric field

intensity

^{in vacuum.}

Then,

for

(qO)Nd = (qO)C02

the relation between the relevant accelerations reads

while the accelerations due to the thermokinetic pressure scales as

Hence,

similar results can be

expected

^to be obtained

irradiating

^the

target by Âo

⁼ 1.06 gm with

(qo)Nd

⁼ R’o and

by Âo

⁼ 10.6 gm but with

(qo)co2 =

^0.01 qo.

This has been confirmed

by

^the simulations and is illustrated in

figure

^{6. Also the initial}

ponderomotive

force induced

compression

times ratio

zNd/ico2

^scales

as

(8’),

^which

might

^be

important

^{in foil}

experiments and/or

for ultrashort laser

pulses.

When the

ponderomotive

^{force is not taken into} account, ^the

electromagnetic

^wave

penetrating

^into

the

plasma

^and

reflecting

^{at the} cutoff forms a

standing

wave with non-zero nodes due to the

absorption

ⁱⁿ

an under-dense

plasma

^{with the last, in a direction}

from laser to target, ^maximum

of 1 E 12 being

^the

absolute one 1 E l’ ab r,.

^max and reached at p =ps Pc

[ 1 7] .

For

higher

^{densities p} >

Ps, 1 E J2

goes ^{to zero} asymp-

totically passing through

^the critical surface.

However, including

^the

ponderomotive

^{force in}

gasdynamics,

the

electromagnetic

field infiltrated

through

^the upper shelf

density region keeping

^its

oscillatory

^character

is then

aperiodically damped, passing

the last critical surface

(see figure 7).

Nevertheless, ^{the results obtained}

FIG. 6. ^- Comparison ^{of the} density p and velocity ^u profiles ^for CD2 target irradiated by ^Nd (go ^{= 1014 W} cm-2) ^and C02 (qo ^{= 1012 W} cm-2) lasers at a few wavelength Â-o ⁼ 1.06 ktm,

Âo ^{= 10.6} gm cutoff (z ⁼ zj neighbourhood, respectively. 1) p(z), Nd ; 2) p(z), Nd, PMF ; 3) u(z), Nd ; 4) u(z), Nd, PMF ; 5) p(z),

CO2 ; 6) p(z), C02, PMF ; 7) u(z), CO2 ; 8) u(z), CO2, ^PMF.

FIG. 7. ^- Field 1 E l’ ^and density spatial profiles illustrating ^the

presence of a sonic point ^{at the} density step ^and modulation of upper

shelf density.

still remain in a

qualitative

agreement with those of Lee et al.

[7] just setting 1 Er ,1’ = JEJ’

^{max and the}

sonic

point

to p ⁼ ps. Transition

through

the subsonic to

supersonic

^{flow with}

respect

^{to the frame}

moving

with the

density

^{step can be seen on}

^figure

^{7 as a}

transition from the coincidence

of 1 E l’

^{maxima and}

density

minima for _p > p.

and 1 E 12

maxima with

density

^{maxima at} the lower shelf

density

p ps.

Acknowledgment.

^- The authors are indebted to Drs. Yu. V.

Afanasyev

and V. B. Rozanov for valuable discussions.

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