high intensity laser plasma interaction.pdf

7/30/2019 High Intensity Laser Plasma Interaction.pdf

1/102

HIGH INTENSITY LASER PLASMAINTERACTION

J.L. BOBIN

Universit Pierre etMarie Curie (Paris 6), Laboratoire de Physique et Opaque Corpusculaires,Tour 1 2 E.5 4 , Place Jussieu, 75230 Paris Cedex 05, France

1NORTH-HOLLAND-AMSTERDAM


2/102

PHYSICS REPORTS (Review Section of Physics 1_etters) 1 2 2 , No. 4 (1985) 1 7 3 2 7 4 . North-Holland, Amsterdam

HIGH INTENSITY LASER PLASMA INTERACTION

J.L. BOBIN

Unirersit Pierre ci Marie (uric (Paris 6), Laboratoire dePhysique ci Optique (orpuscu/alres,

Tour 12 ES. 4. PlaceJussieu, 75230 Paris Cedex ((5~Frnace

Received November 984

Contents:

I. Introduction 1 7 5 III. Wave Couplings 2262 . Light absorption and plasma motion 7 5 21 . Purely growing modes and 3-wave instabilities 22 6

2 2 . Instability dynamics 22 9

1 . The Linear Regime 7 7 2 3 . Influence of the pump tuning and intensity 23 53 . Equations of the linear regime 1 7 7 24 . Pump depletion 2 3 7

4 . Dimensional analysis for collisional absorption 1 7 9 25 . Coherence and incoherence 2 4 1

5 . Dimensional analysis for plasma flow dynamics 1 8 ) ) 26 . Scattering instabilities in homogeneous a nd in-

6. Collisional absorption as a function of the incident inhomogeneous plasmas 2 42

tensity 1 8 2 27 . Filamentation revisited 24 7

7. Structure of the conduction zone: classical transport 1 8 3 28 . Laser interaction with an inhomogeneous plasma flow 2 51

8 . Long pulses, short pulses 1 8 8 29 . Wave breaking 26 3

9 . Laser driven detonation 1 8 9

I)). Limits of the linear regime 1 9 1IV . Harmonic Generation 2530 . v X Bnon-linearities 2 55

II . Effects ofthe Ponderomotive Force 9 331 . Oscillating mirrors 2 57

II. Ponderomotive force and pressure 1 9 31 2 Resonant propagation and absorption 1 95 32 . Raman upeonversion. Couplings and cascades 25 9

1 3 Electron bunchingCavities 1 99 33 . Growth rates and thresholds 26 31 4. Steady structures, profile steepening 2 01 34 . Spectrum of harmonic lines 264

35. Intensities of harmonic lines 2 681 5. Isothermal regimes 20 61 6. Anomalous transport 2 1 ) 7

1 7 . Possible physical origins of flux limitation 21 0 3 6 Conclusion 2 7 )1 8 . Flux limitation form transport theory 21 3 References 2 7 1

1 9. Filamentation and self focusing 2)7

2 )) . Selfgenerated magnetic fields 22 )

Single ordersforthis issue

PHYSICS REPORTS (Review Section of Physics Letters) 1 2 2 . No. 4 (1985) 173-274.

Copies of this issue may he obtained at the price given below. A ll orders should he sent directly to the Publisher. Orders must he

accompanied by check.

Single issue price Dfl. 66.00, postage included.

0 370-l573/85/$35.70 Elsevier Science Publishers By. (North-Holland Physics Publishing Division)


3/102

J.L. Bobin, High intensity laserplasma interaction 1 7 5

Abstrac t:

Basic mechanisms, linear a n d non-linear, are reviewed. Light absorption may take place linearly through inverse bremsstrahlung. Together with

usual heat transport by electrons it leads to a linear gas dynamical regime whose main formulas are given. When absorption occurs non-linearly, byresonance in a density gradient or through wavewave couplings, several non-linear regimes may show up . They are investigated in connection with

the ponderomotive force, soliton formation, wavebreaking, , , , The processes responsible for harmonic generation are given special developments.

1 . Introduction

When a high intensity laser beam is focused onto a solid surface, matter is strongly heated. The

temperature is so large that a dense plasma is formed and set into motion. As pointed out long ago[13],the process is of interest for controlled thermonuclear fusion. Subsequent developments led to the

concept of laser driven implosions [4]. Obviously the physics of laserplasma interaction are of

paramount importance: the feasability and the efficiency of the proposed fusion devices depend on the

way energy is transferred from the light to the plasma.One is then induced to ask the following questions:

How does the plasma absorb radiation?

What is its thermodynamical state?

Which forces act on it?What is its motion?

Answers to these questions are not independent from one another. Many unknowns still remain after

some twenty years of extensive research. This is due to the complexity of the basic mechanisms whichfurthermore are strongly intermingled.

However, a separation naturally occurs between linear and non-linear regimes. The main features of

the former were determined some 12 years ago [58].The latter are still under investigation. They

belong to the more general field of plasma non-linearities which is rapidly growing and changing. In the

case of laser interaction three main parts may be distinguished: effects of the ponderomotive force,wavewave couplings and harmonic generation. They can all be related to specific non-linear terms in

the coupled equations describing the fields and the plasma.

The present review is organized accordingly. The viewpoint is rather theoretical: fields obeyMaxwells equation whilst the plasma is described fluid dynamically. The equations are presented first.Then, four chapters deal with the linear regime and the three classes of non-linear phenomenaintroduced above. Most of the paper deals with a qualitative picture with computational and experi-mental checks.

Recently, due both to theoretical considerations and to improvements in frequency conversion, a

special emphasis was put on the wavelength dependence of laser interaction in the visible and near U.V.parts of the electromagnetic spectrum. The Nd-glass versus CO

2 radiation is an old debate, not

completely settled. It is generally admitted that lower frequencies favour non-linearities (the well-

known Lk2 scaling) and accordingly lower the absorption. The gas dynamical behaviour becomes less

controlable, a major drawback for those interested in inertially confined thermonuclear devices.

However the details of the wavelength dependence are still poorly known, except in the linear regime.

2. Light absorption an d plasma motion

A laser created plasma is electrically neutral but for local fluctuations. An intense oscillatingelectromagnetic field acts primarily and directly on the far less massive electrons. Action onto the ions

mainly takes place through locally induced space charge fields.


4/102

1 7 6 J.L. Bobin, High intensity laserplas ma interaction

Let EL and BL be the electric and magnetic fields, respectively, of the laser wave. Let E5 be the spacecharge electrostatic field. The electrons have a velocity v, a particle density n, exert a pressure Pc andundergo collisions with an effective frequency ii. The dynamics of the electrons are described by an

equation of motion:

( 9 v VPem[-_+(v.V)] v = _e(Es+EL+XBL)-- miv (2.1)

coupled to a wave propagation equation deduced from Maxwells equations~2 I?

UL..L 22

+ c V EL = 4ire nv.a t

2 a t

E5 is related to charge density fluctuations through the Poisson equation

V~E~41T(~n)e. (2.3)

Three sources of non-linearities are clearly visible:

v a(v V) v, x BL , nv. (2.4)

c

The first one is at the origin of a non-linear force [9, 10]. The second results in the generation of radiation

whose frequencies are the harmonics of the incoming one, see e.g. [11]. The third term represents

wavewave couplings [12, 13].The non-linear terms may be neglected through the usual linearization procedure. Assume that v and

n, both have two components: a slowly varying one, u , n 0 plus a high frequency perturbation , .

Furthermore, terms of order v/c are neglected. The high frequency part of (2.1) reduces to

ai~/at+vii eE/m. (2.5)

When coupled to (2.2) linearized i.e.

( 92E L /(9 t2 + C 2 V 2 E L = 4iren

0 ( 9 i i / ( 9 t , (2.6)

it allows the straightforward calculation of an energy absorption coefficient K r, ,: for w greater than Wp

K~= cw2(1 ~2/~2)112 (2.7)

and for w close to w,

K~,= (2w~v)t2/c (2.8)

where v~,is the plasma frequency


5/102

J.L . Bobin, High intensity laserpla sma interaction 1 7 7

= 4zrn0e2/m. (2.9)

The fluid dynamical derivation of K~is actually incomplete. In order to deal with the whole of therelevant physics, it is necessary to start from a kinetic equation. This w as done in the case of theVlasov Maxwell system of equations [14, 15]. The absorption is calculated in presence of discrete fixedions. The result is a high frequency conductivity which includes a factor:

exp{ik (r, ri)}) (2.10)

i.e. the spatial correlations of the ions. Here r , denotes the location of ion i. The absorption is differentfor randomly distributed ions or for ions whose spatial distribution is consistent with correlations at

equilibrium. This factor is usually overlooked when computing laser driven implosions. However whenthe target is made of high Zmaterial the ion plasma parameter (Tbeing the temperature in energyunits)

F1 = Z

2e 2(n1)

113/T (2.11)

can be of order 1 , even at electron temperatures of several hundreds keV, indicating strong ion

correlations which could contribute to an enhanced absorption. Since in the following, absorption willbe considered as an energetic boundary condition for the flow, the fluid approach is to be taken up.

Now, the low frequency part of (2.1) is

(9u/3t+(uV)u= eEs/m Vp~/nom ~0 (2.12)

since the slowly varying velocity u , i.e. the flow velocity, is much smaller than the electron thermal

velocities, it can be neglected together with its derivatives. Then both sides of (2.12) are approximately

zero, thus yielding a Boltzmannian distribution for the electrons. On the other hand, the equation ofmotion for the ions with mass M and charge Z is

0u / (9 t + (u . V) u = ZeEs/MZVp1/noM. (2.13)

Eliminating E~between (2.12) and (2.13), one gets an equation of motion for the neutral plasma as a

whole

3u/ (9 t + (u V ) u = ~ZV(pe +p1)/n oM= Vp/p (2.14)

where p is the fluid mass density and p the total pressure, ions plus electrons.

I. The Linear Regime

3. Equations for the linear regime

When the oscillatory velocity is much smaller than the thermal velocity of the electrons, the

electronion collision frequency i- is a function of the electron temperature T~(in energy units). It readsafter Scheuer [16]

2 ZC4fle

3 m~(21TTe)3~In A (3.1)


6/102

75 J.L. Bohin. High intensity laser plasma interaction

where the Coulomb logarithm deals with the ratio

3( T)31 2.1 = (3.2)2Ze3(~n~)2

of the Dehye length to the impact parameter for a 9 0 0 collision. Substituting (3.1) into (2.5) yields a fieldamplitude independent absorption coefficienl K,,,. The linear collisional absorption of light is readilyincluded in the energy equation in the form

~ (p ~+ pU) + ~. u(p ~+ pU +p) +I+F) =0 (3.3)

in which U is a specific internal energy, F a heat flow, and Ithe absorbed intensity such that denotingby I its value outside the plasma and by ~ the radiative intensity:

I=I~4 with V.~=K,,,. (3.4)

The set of equations (2.14). (3.3), (3.4) should he supplemented by the continuity equation

Ilp/9t +V~(pu) =~. (3.5)

One thus gets a complete description of the fluid dynamical behaviour of a single component fluid

(plasma), irradiated by a laser beam and linearly absorbing.

Such a system of equations cannot be analytically solved in general and requires the use of numerical

methods. Computer codes were devised and widely used mainly for calculating laser driven im-

plosions: Medusa [17], Lasnex [18]. However, analytical solutions ca n be found in some special

instances: self similar flows or even simpler, steady one-dimensional situations. Indeed, in the lattercase, one gets the following first integrals in a reference frame moving with the structure under

investigation

Pu =J . u =JV, (V= lip) ( 3 . 6 )

J2V+p =J2V+p (3.7)

/J2V2 y J2V~ y~ (3.8)

2 yl y~l

Here an ideal fluid is assumed. J is a constant mass flux. The state labelled 0 is a uniform subsonic

radiation-free reference state. Its density is of course greater than the critical value p. for which w=

m AlPci~ (39)4n~eZ

Equations (3.6)(3.8) can be solved for any interesting quantity. However if one is not interested in the


7/102

J.L . Bobin, High intensity laserpla sma interaction 1 7 9

details of the structure, much of the relevant physics can be unraveled from dimensional considerations,

a common feature in fluid dynamics.

4 . Dimensional analysis for collisional absorption

Assume a laser beam is impinging onto a very gentle plasma density gradient: the characteristic

length L is much longer than any absorption length 1 . The underdense absorbing layer has an almostconstant density p


8/102

1 8 0 fL. Bobin, High intensity laser plasma interaction

and independently of a and t

p I213p3. (4.7)

The second equation (4.6) indicates that p increases with time, i.e. the absorption zone in theinhomogeneous profile, progressively moves towards higher densities, as time elapses. Moreover, the

length 1 of the zone also increases according to

1 a11~I~6t9~6. (4.8)

The same behaviour occurs for the temperature since from (4.5) one also has

T ah/419/8tts/s . (49)

Thus, the existence of the self regulating regime is limited in time. Since monochromatic radiation

cannot propagate beyond Pc, as soon as the absorbing zone reaches the vicinity of Pc, another regimenecessarily takes place. Now (4.7) tells us that for given I and Pc, p has a fixed value. Due to the

equation of state (of the ideal gas type e.g.)

Z+ 1

Mp~T (4.10)

where Z is the average charge of ions with an average mass MTno longer varies either.Combining then (4.7) and (4.10), one gets a relationship independent from a and 1 , i.e. from any

assumption about the absorption mechanisms. It suffices to state that the absorption takes place at the

cut-off density, so that

T (I/pc)213. (4.11)

Now, one may use (3.9) to link p and T to radiative quantities only: intensity and wavelength

A 2i~c/w.One thus gets

T ( IA 2)21 3 ~. (4 12 )p (I/A)~3~I

when collecting experimental results from the literature (e.g. [19]) it turns out that for IA 2 greater than1012, Tis proportional to ( IA 2)21 3 . Fitting the formulas (4.12) to measurements, and putting Tin keV, Iin W/cm2, A in ~imand p in megabars, yields

T= 109(1A2)213 (4.13)

p = 1.4x i0~(1+ 1/Z)(I/A)213. (4.14)

5. Dimensional analysis for plasma flow dynamics

No motion was considered in the preceding section. In order to introduce a flow velocity into the

dimensional analysis, one starts from the dimensions ofp and I:


9/102

J.L. Bobin, High intensity laser plasma interaction 1 8 1

[p][L][t]_l = [I] (5.1)

i.e., there exists a characteristic velocity linking the pressure obtained in the plasma to the absorbedintensity.

No w when a plasma flow is driven by a laser beam impinging onto a solid surface (or less commonly

onto a high density gas jet [20])part of the flow is unsteady: a rarefaction wave sw eeps out a low densityplasma. A stable transition to a stationary zone is possible provided the particle velocity is the localsound velocity:

c~=[(Z+ 1)T/M]H2 = (p /p)U2. ( 5 . 2 )

Therefore when absorption takes place near a sonic point, one gets from (5.1) and ( 5 . 2 ) the propor-tionality

p3 12p~1 12c c I (5.3)

already obtained in (4.7) for a non-moving plasma. When the sonic point is at the critical density, eqs.(4.12) are recovered. Still no assumptions are made concerning absorption mechanisms. One may theninterpret the empirical law

T (1 A 2 )2 1 3 (5.4)

as denoting absorption at critical density which is also a sonic point for the flow. This is valid for any

absorption process linear or non-linear.

The validity domain of (5.4) is fairly well known from experiments in which I and Te can be

measured accurately enough. In the plane ln T , ln(1A2) of fig. 5.1 experimental points lie within a factorof 2 around a skeleton which includes a segment with slope 2/3. Everyone agrees with: (5.4) holds forthe range

1012


10/102

1 8 2 iL. Bobin, High intensity laserpla sma interaction

1 0 1 5/A 2 although different power laws were put forward specially at Livermore [22]

Te (IA2)~5. (5.5)

The cold component exhibits a much slower variation which is poorly known.When (5.4) is not satisfied, this may be due to tw o reasons: either the collisional absorption does not

take place at a well-defined density, an unsteady situation for the whole duration of the laser pulse, orthe sonic point is not where absorption is. Both effects are likely to occur simultaneously for the lower

intensities (1A2< 1012 Wp.2/cm2). Then the plasma flow is driven by a volume absorption in the

rarefaction at densities well below critical. For the highest fluxes (IA2> 1 0 1 4 Wti.2/cm2) the situation is

not that clear. Indeed a competition sets up between instabilities due to the non-linear coupling ofelectromagnetic waves with plasma modes, and ponderomotive effects which tend to steepen the density

profile.

6. Collisional absorption as a function of the incident intensity

In the previous sections, scaling laws were set up. They al l involve the absorbed intensity I . This isonly part of the incoming intensity I~.The connection between the plasma parameters such as thetemperature and I~required the quantitative knowledge of the absorption processes and the subsequentflow dynamics. In the linear regime, one may assume that the absorbing underdense zone is an

exponential rarefaction

p=p~e~ (6.1)

where L is a time dependent characteristic length

L=c.,t. (6.2)

The plasma flow is always slower than electromagnetic wave propagation. Accordingly, one may solve

the transfer equation (3.4) at any instant. Now the incoming light propagates in the underdense plasmaup to Pc. At that point it undergoes a mirror reflection and then propagates backwards. Along most of

the profile the absorption coefficient is given by eq. (2.5). The exception is the vicinity of the critical

density. However it turns out that one does not make an important error by using (2.7) for the whole

density profile. Then the ordinary differential equation

d4 = AA2p~e_sx/L (6.3)

dx T312(l e~~)2

is separable. Its solution with the boundary condition: ~ is I~at infinite x . isIn ~ = 8AA2p

0L (6.4)

~ 3 T3 1 2

Now, one may rewrite (4.13) as


11/102

J.L . Bobin, High intensity laser plasma interaction 183

T312/A2= BI. (6.5)

Furthermore Pc is proportional to A2. One thus gets

~ (6.6)

J o IA4

In (6.4), (6.5), (6.6) the quantities A,B and C are constants. The result (6.6) is better expressed in termsof an absorption efficiency

= I/Ia (6.7)

as

flA ln(1 7)A) = CL/10A

4 (6.8)

i.e., the absorption scales as I0A

4. Figure 6.1 displays i~AversusIas it results from numerical computations

(A = 1 ~m) [23].

1 01 3 10~1~ \,.J/cm2Fig. 6.1. Absorption efficiency versus Ifor a CH plasma: (a ) L=50 ~sm,(b) L = 100 tim, (c) L=50 0 tim.

Absorption increases strongly with L. At large L, the incoming intensity does not even reach the

cut-off density: volume absorption. At high radiation fluxes and small L, ~A goes to zero but the linear

model no longer holds. Indeed non-linear mechanisms take place and tend to modify the fractionalabsorption.

Although simplistic, and not too realistic, the model used in this section allows one to draw some

general conclusions:

A significant scaling is 10 A

4 .

The transition between a situation with a strong collisional absorption and the one with a small

absorption takes place over about tw o decades of the significant parameter 10A

4.Due to the presence of L in (6.8), plasma dynamics might play an important part. Actually, it turns

out that its main influence deals with the scaling. One finds that, in general, ~A scales as 10A s where S

(>4) is an exponent depending upon the plasma dynamical regime. For instance in a self similar

situation, S is 5 as shown by Mora [24].

7 . Structure of the conduction zone: classical transport

We now are interested in the overdense region (p>Pc) in which laser radiation does not enter. Its

structure is dominated by the thermal transport term F which appears in (3.2). The so-called classicalor normal transport term was derived long ago by e.g. Spitzer Harm [25] and Braginskii [26]. This is


12/102

184 IL. Bobin, High intensity laserpla sma interaction

the one usually given in plasma textbooks. It reads

F=~T512VT (7.1)

where x is a constant. It refers to steady state situations with moderate gradients and energy fluxes.These conditions are not always fulfilled in laser plasma interaction, a point to be discussed later (see

section 10).In a stationary regime, the energy equation in one-dimensional geometry is

a r iu2 y \ 1pu(_+_c~)+FJ=0. (7.2)

ar~ \2 yl

For a fully ionized plasma y is 5/3 and one immediately gets a first integral

u T A1 )

p[~D?2+5IKT312~- (7.3)

2 ar c.

where

V~=u/c. (7.4)

is the local Mach number, while referring to (7.1) and (5.2)

K=~M/(Z+1). (7.5)

A0 is the integration constant. It represents the energy exchange between the conduction zone and the

external world: laser radiation flux converted into thermal flux at cut-off. N ow Ao/c~is of the form

JV0/V(< < J ). Thus it is negligible in most of the structure. When the flow is fairly subsonic (~j~2~g5) thefactor (~2+ 5) in (7.5) is approximately constant. In the plane case

pu=J, A0/c~=~0, (7.6)

the equation is readily integrated. The solution with suitable boundary conditions is

T= Tc(1 x / x o )2 1 5 (7.7)

where T c is the temperature at cut-offand x0 is the thickness of the conduction zone

4K T~

2 T~

(7.8)25 ft Pc

separating the critical density from an ablation front where

T0, aT/ax~x. (7.9)


13/102

IL. Bobin, High intensity laser plasma interaction 1 8 5

In the spherical case

J= 4irr~pu (7.10)

and one finds

2/5 2/5

T=T~( ~ ) (i_~) (7.11)T~ T o T

in which r~and r0 ~ r~ ,are the radii of the critical layer and the ablation front respectively. One has

25J

= 16~rKT~2 (7.12)

and r~has a rather complicated expression we may omit.The well-known temperature profiles for the plane and the spherical case are displayed on fig. 7.1.

x~ x0 ~

a) b)

Fig. 7.1. Temperature profiles: (a ) plane case, (b) spherical case.

In order to complete the profile determination in plane geometry, one uses the first integral of the

equation of motion (3.6) which is rewritten

P/p+p=Ju~+p~. (7.13)

The pressure is then a linear function of the specific volume V= i/p. Since the point at cut-offis sonic,

Ju~=p~, (7.14)

and accordingly

= (~1R~+ 1) if one has ~V?~~ 1, (7.15)

where ~V?0is the Mach number for the ablation front. It is generally negligible so that the ablation

pressure is twice the pressure which prevails at cut-off. This pressure is much greater (usually megabars)


14/102

1 8 6 fL. Bobin, High intensity laserplasma interaction

T~ [_~ P

Sonic

>~__~-=


15/102

IL. Bobin, H igh intensity laserpla sma interaction 1 8 7

1/20

(7.20)T~ \~s/ A

For the 1.06 p~mradiation of the Nd glass laser, this efficiency is about 0.1 (Fauquignon and Floux [6]).For the spherical geometry, the profiles can be determined after the differential equation of motion,

which in its simplest form, reads

pu du = dp. (7.21)

Using an ideal gas equation of state and the definition (7.4) of the Mach number, one may transform the

above equation either for u:

dlnu= 1 2dln~ (7.22)

1-~P?

or for ~P?:dln~V~+~dlnT= 1 2dln~. (7.23)

1-~P?

From (7.22) it is readily deduced that for small~V?,u hasa local maximum at r = (3/2)1/2 r o and in this vicinity

Tx r2uccl/p. (7.24)

From (7.23) one gets the behaviour of ~ which can be entirely calculated, at least numerically, after

the knowledge of T, either in the overdense region (7.10) or assuming an isothermal plasma in the

underdense zone. At large ~JJ~(T= T~)one gets

= 2 ln(r2/T~)c c u2. (7.25)

Thus, the whole structure, including the underdense rarefaction has a stationary structure according to

the scheme of fig. 7.3.

In al l the above calculations, ions and electrons have one and the same temperature TActually,laser light heats the electrons which in turn transfer their energy to the ions through the usual relaxation

term

r~J~t

Fig. 7.3. Steady flow structure in 1-D spherical geometry.


16/102

1 8 8 IL. Bobin, High intensity laserplasma interaction

dTe TT, mM /T T\312Teq ~+) . (7.26)

dt Teq fleZe lnA m M

The temperature decoupling can be accounted for [71with the result that in steady state, the electrontemperature is only slightly higher than the ion temperature. At cut-off, one finds

Te~1.5Ti. (7.27)

8. Long pulses, short pulses

Equation (7.8) gives the thickness of the conduction zone in plane one-dimensional geometry. Using

the mass number A of the plasma ions it may be rewritten

1/2

x0 = 300 (---) TC

2A2 (8.1)

where x0 and A are in micrometers and T c in keV. The numerical coefficient w as chosen in order to fit

numerical simulations [28]. Obviously, time is needed to build up a zone with such a thickness. The

order of magnitude for the minimum required time is

tc = 1.3 X 109T312A2 = 4.5 X 10231A4 (8.2)

where t is in seconds, Tin keV, Iin Wcm2 and A in micrometers. The sound velocity is

1/2

= ~ ( A ) 10~T~2cm/s. (8.3)

The build-up time for the conduction zone is thus independent of the target material. For a givenabsorbed intensity it increases as the fourth power of the wavelength. It is impossible then to reach a

steady state albeit localized in the flow for a shorter pulse.Putting numbers in the formulas, for an absorbed intensity 1013 W/cm2, tc is 4.5 X 10_lOs for radiation

with wavelength 1.06 ~m (N d glass laser), 2.8 x 10- ~ s for radiation with wavelength 0.53 ~im(frequency

doubling of Nd laser light), 4.5 x 1 0_ 6 s for radiation with wavelength 10 ~m (molecular CO2 laser). A 10 0

picosecond pulse is long for radiation at 0.53 p.m, short for the two otherwavelengths. Experimentsconfirmthis statement as shown on fig. 8.1, on which flA is plotted as a function of the scaling parameter IA

4 ofsection 6. The wavelength and pulse durationsare those used by Amiranoffet al. [29].The steady regime is

obtained either for 80 psor for 2.5 ns at 0.53 p .m (sameLaccording to fig. 6.1). On the contrary at 1.06 p.m, Lis much smaller for iOops than for 2.5 ns, the steady state being reached in the latter case only.

One may also notice that by comparison with the model dealt with in section 6, the maximumfractional absorption is smaller for shorter pulses. The difference comes from the choice of (2.7) for the

absorption coefficient. Had we taken up (2.8) together with the same exponential density profile, thetransfer equation


17/102

IL. Bobin, High intensity laser plasma interaction 18 9

1

--

=4T2ns

10 X

4

0 ~~4a/cm21 01 1

Fig. 8.1. Scaling of experimental absorption efficiencies.

d4~/dx= ap~2e~~2~/ (8.4)

would have led to the solution

l = 1 exp( ~aLp~4)= 1 exp(bL /A312) (8.5)

where a and b are constants. The correction exp(bL/A32) is larger for smaller ratios L/A3~2.Given A,it will be larger for short pulses since Lhas no time to reach its steady state value. Note that the way

(8.5) was obtained is valid for short pulses only: steep gradient, absorption at cut-off, negligible in theunderdense plasma.

With low intensities associated with long pulses, the non-total absorption comes from partial

reflections along the density gradient (Fresnels formulas, or B.K.W. analysis) none of the previous

calculations accounted for.

As far as absorption takes place mainly at cut-off, a sonic point, the scaling laws (4.12) are not

affected by the steady or unsteady behaviour of the flow in critical and overdense regions. Since, on theother hand for short wavelength long pulse intensities smaller that the instability threshold, absorptionis comparatively high and weakly depends upon the intensity, it is not surprising that an experimental

scaling law as (10A

2)213 be fairly well satisfied for the temperature. Astonishingly enough it also holds for

1 0 p .m radiation in short pulses (low absorption). This could be due to effects represented by (8.5) with

the consequence that the curve lA(J0A4) is very flat around 50%, as shown on fig. 8.1 for 10 0 ps pulses

of 1 p.m light.

9 . Laser driven detonation

So far, it was assumed that the density profile encompasses overdense regions. This corresponds tolaser radiation in the visible part of the electromagnetic spectrum impinging on to a solid surface. In

other instances such as gas breakdown, no overdense medium is present. This is the main differencewith the cases dealt with in the previous sections. Since laser light absorption still creates a high

pressure plasma, a shock wave is a necessary transition to the unperturbed material. Now, therestrictions of thermodynamical origin which prevent the shockand the absorbing layer to propagate at


18/102

l9)) iL. Bobin. High intensity laserplas ma interaction

one and the same speed, no longer hold. Then a stationary structure with the same mass flux i through

both fronts may build up. The situation is thus close to the one which prevails for chemical detonation.Assuming that light absorption produces a state in which the particle velocity equals the local sound

velocity (ChapmanJouguet condition) one gets the optical analogue of the Zeldovich [30], Von

Neuman [311,Dring [32] scheme shown on fig. 9.1.

The corresponding temperature and density profiles are shown on fig. 9.2. The light may come fromeither side of the structure.

\Shock adiaba~(Hugoni&)

~ Isen~rope

\ \ \ //

9V

V 0 Vcj V 9 Absorphon

F i g . 9 . 1 . Pressure/specific volume diagram for the optical detonation: F i g . 9 . 2 . Temperature an d density profile in th e optical detonation.

initial conditions p o V 0 , shock conditions p, V~, final s t a t e p~jV~J.Absorption takes place between V 0 a n d Vcj.

In laser induced gas breakdown (for a review see e.g. Dc Michelis [33], Grey Morgan [34].

Ostrovskaya and Zeidel [351, Raizer [361),on e usually observes such a detonation front propagating

towards the focusing lens. The absorption is then almost total. When this is not the case a second frontis also seen which propagates away from the lens. For instance, in an experiment by Gravel et al. [37]

using a CO2 laser, the breakdown spark is driven by a first high power pulse of about 10 0 ns F.W.H.M.

(Full Width Half Maximum). The spark is a diverging lens for the radiation in the subsequentmicrosecond low power tail of the pulse. This radiation is focused beyond the geometrical focus,

interacts with the real boundary of the spark, and finally drives a fast detonation propagating away fromthe lens.

The propagation velocity of the laser supported detonation (LSD) is readily calculated, provided the

ChapmanJouguet condition holds, i.e. in the final state

U~j= YPcjVcj. (9.1)

One then finds, Ramsden and Savic [38],

D= [2(y2l)I/pg]US (9.2)

where p5 is the density of the unperturbed gas. Similarly one gets the hot plasma temperature.

Champetier [391,


19/102

IL. Bobin, High intensity laserplasm a interaction 1 91

T~~=1---~~ (9.3)

1+a 2 (y+1)2

where a is the degree of ionization. One finds again a j2/3 dependence of the temperature. But as far as

the whole structure is underdense, there is no influence of the laser wavelength.

10 . Limits of the linear regime

When performing experiments with increasing intensities on targets, one finds a transition around 2

to 3 x 1 01 3 Wp.2/cm2, then as shown in fig. 5.1, tw o electron temperatures show up. The lower is almostconstant around 500eV [21], the higher still obeys the scaling law (4.11). Furthermore light absorptionturns out to be higher than predicted by the linear theory [40,41] . The actual behaviour is displayed on

fig. 10.1.

111A

0 ____ _____

Fig. 10.1. Fractional absorption versus intensity: experiments (hatched area) and linear theory.

Obviously beyond ( 2 3 ) x10~~Wp.2/cm2, the linear theory, as presented in the previous sections, no

longer holds.Now, in section 2 , 3 non-linear terms were identified in the equation. They all involve the electron

velocity. In alaser irradiated plasma, the electron velocity has two components:

an oscillatory part

=eE/wm (10.1)

where E is the electric field of the wave,

a thermal part whose distribution and mean values are known provided a temperature does exist

Ve = (V) (T/m)~2. (10.2)

In the linear regime, it is assumed that Ve is dominant and, accordingly, 5 is neglected. Non-linear

processes are expected to be important wherever the ratio 5/Ve is no longer small. Hence an a priori

requirement

52 = e2E2 = ~J~A~>1 (10.3)

v~w2mT ircT

where the following relationships were used


20/102

19 2 fL. Bobin. High intensity laserplas ma interaction

= cE2/4i~, r

1 1 = e2/mc2 (classical radius of the electron) . (10.4)

In (10.3) a scaling law in 10 A

2 is again found.

The linear absorption coefficient is actually

K~. , ,= ae2/v3 (10.5)

where v is the electron velocity. So far it was identified with the thermal velocity, hence the T 3 1 2 in the

denominator. For high irradiances one may rather substitute L ~ instead of Ve. One then gets the

proportionality

K0~ , j~3/2, (10.6)

i.e., a strongly decreasing absorption. As shown on fig. 10.1, this is not the case proving that

mechanisms, other than collisional, contribute to light absorption.

Another feature of the linear regime comes from neglecting radiative effects: on the one hand, the

pressure and the energy density associated with the impinging high intensity laser beam; on the otherhand, radiative losses ~ due to thermal emission. Now these processes should be accounted for in the

more accurate equation of motion and the energy balance equation. In order to discuss the order ofmagnitudes, it suffices to restrict oneself to steady situations. In plane. l-D geometry the energy

equation without radiative terms can be rearranged as follows [71:

J /y+l 1 1 ~ /y+ I I l\ 1I(~~)+F= L - - ) ~ ~ )po I. (10.7)2~Y

1P P a Y1Po P ~1

When including the energy density and the pressure due to the laser radiation, one gets in the

right-hand side of (10.7) the supplementary term:

J 1 1 2EL J1 0 a

(10.8)2 fJ P o Pc Pc C

since both EL and PL are of the order of 10/c. This term will be important if u 10/c is comparable withthe left-hand side of (10.7). This is possible if

i) either

+F ~ I~ (10.9)

i.e., low absorption and transport in presence of a high incoming intensity. This situation is also

obtained if, even with absorption, most of the energy is reradiated so that a small fraction only:I ~ Ia), actually drives the flow. Thermal radiation emission increases with the average chargenumber (Z) of the plasma ions. Hence the experimental observation of enhanced non-linear effects whenshining a given laser intensity on to a higher Z target [42],

ii) or

I+F-~-I() (10.10)


21/102


in a relativistic flow. Since no such flow occurs in actual laser interaction experiments, this case will be

henceforth discarded.

II . Effects of the Ponderomotive Force

11 . Ponderomotive force an d pressure

They result from averages over a period of an oscillating field. The electron equation of motion

should include non-linear terms. Only the friction term in (2.1) is to be neglected. One then

obtains a complete equation for the low frequency components (u , n0, E, B) by averaging the high

frequency component (5, n , E, B) viz.

m ~= m(u .V)u e(E+~xii)_!~_ m((5-V)S) e (~x~)+-~(nV); (11.1)

here

PefloT (11.2)

is the mean electron pressure. Obviously the three first terms in the right-hand side of (11.1) do notdepend on a high frequency (laser) field. On the contrary the three last terms are non-vanishingaverages of the non-linear parts of (2.1). They represent a low frequency non-linear forcef~which isreadily calculated after linearized equations for the high frequency components, i.e.

9 t m mn0(11.3)

- mcB=-VxS

e

Substitution into (11.1) yields

f~_m[((5.V)S)+ (SX Vx 5)_~(V)]. (11.4)

For an electromagnetic wave in a uniform plasma the last term within the brackets vanishes.

Furthermore in general n/no is small so that even if the oscillating kinetic energy and its gradient are not

large with respect to the thermal energy and its gradient, the main contribution tof~comes from thetwo first terms within the brackets. Finally for electrons one gets after some elementary vector calculusF~=nof~=mnoV~-

2=--~ (11.5)

2 2mw w 81r Pc 81T

This force clearly derives from a potential which is the radiation pressure associated with the high


22/102

1 94 fL. Bobin, High intensity laserplasma interaction

frequency wave. We then write

F0= dpL/dx (11.6)

in which, for an inhomogeneous plasma

p itp (E

2) I f(E2)PL~J dp, (11.7)

Pc8~ Pc 81T

P u being the density of the overdense (radiation free) reference state. It behaves the same way for anelectromagnetic (laser) wave or for a longitudinal plasma wave. In the following we are chiefly

interested in the radiation pressure associated with a laser beam interacting with the driven flow.

The ponderomotive force comes in the plasma dynamical equation as an external force. It acts on the

electrons only. However through charge separation fields, it actually acts on the whole plasma, as one

may easily find by the same token as used to set up equation (2.14). The new equation of motion is thus

~+(u~V)u=-~-V(p+pL). (11.8)at

The energy equation is also modified. In a plane steady regime one gets the first integrals in p andV(= l/p)

V+p + p . = J 2 V0 +Pa

____ (11.9)\2 yl 8ir 1 2 yl

Eliminating p between the tw o yields a quadratic equation for V. This ca n be cast in a non-dimensionalform by setting (y =

=p/po= V0/V, ~P1~= ~J2Vop

0

PL 0 I+F ~E2) Vc (11.10)

W~ l 5 J p o V 5 u 8 i r ( 1 5 p o V ~ )

w and 0 are not independent. They are linked by a relationship which depends on the interaction withthe plasma. One need not enter into details here. In situations where absorption takes place (I + F)/J is

negative so that 0 is locally positive for realistic interaction as shown in [431. The quadratic equation

(11.11)

can be discussed with respect to the parameters to and 0 (>0). In the conditions corresponding to theimpact of a laser beam onto a solid surface, the solution ~t(~JJ~)has the behaviour shown on fig. 11.1.

We are interested in the ~.t> 1 branch only. The solution exists if and only if the mach number ~D?~in


23/102

IL. B o b in , High intensity laser plasma interaction 1 95

~subsonic ~ sonicI V

~Y supersonic/0 ~

1

Fig. 11.1. Specific volume variation across the front versus Mach number in the overdense reference state.

the dense fluid is smaller than some value (


24/102


cVXE=aB/at, cVxB=~aE/at. (12.2)

Now, consider an electromagnetic wave

E= Eexp{i(wt k r)}, B = B exp{i(wt k r)}. (12.3)

It obeys the propagation equations

V2E+ ~~EV(V~E)=() (12.4)

w2 1V2B+~~B+(V~~)xVxB=() (12.5)

C

in which, for an inhomogeneous medium, ~ is space dependent.The simplest case is a linear profile, fig. 12.1: at x = 0 , n = n~and at x = L, n = 0. Then

~(x)=x/L. (12.6)

A plane electromagnetic wave comes from the vacuum side with an angle of incidence 0 . Assume it is

linearly polarized. When the electric field is perpendicular to the plane of incidence (the so-called

S-polarization) one gets standing waves (fig. 12.2) with reflection at the mirror point such that

flflcC0520 (12.7)

The figure represents the energy density in the standing wave. The amplitude increases towards theinside of the plasma. It is mathematically described by an Airy function.

The situation is rather different when E is in the plane of incidence. Indeed there exists alongitudinal component

cLE~=Bstn0 (12.8)

x

.....~

F ig . 12 .1 , Oblique incidence onto a linear density profile.


25/102


(E2) .--~~~n~ (E~

-LOx

Fig. 12.2. (E2> profile for S-polarization. L is much longer than the Fig. 12.3. (E2) profile for P-polarization. The resonance height is

wavelength AL of the light, limited by damping.

where the magnetic field of the wave whose direction is along Oz satisfies the differential equation

d2B dB w2/x(12.9)

dx xdx c L

The solution of this equation is close to an Airy function with the property that B is non-zero for x = 0.Then (12.8) shows a divergentE~for x = 0 (n = n~,~ = 0). Actually E~does not go to infinity since the

waves are damped (absorption). Hence the standing wave profile displayed on fig. 12.3.

The high amplitude oscillation excited at the critical density is purely longitudinal. It results fromtunneling across the electron density gradient. Obviously, this effect takes place if and only if the

electric field has a component along O x. In the case of a linear density gradient the solution can be

calculated analytically. The well-known properties of Airy functions allow one to evaluate for every

abcissa the value of B as a function of the incident wave electric field amplitude E0 (outside of the

plasma, i.e. to the left of fig. 12.3). One then gets approximately:

B(n~c o s2 0) ~0.92Eo(wL/c)6, (12.10)

wL -1/2 2wL

B(n~)0.92 E0 ( ) exp{_ ~ sin

3 o} (12.11)

so that in the vicinity of X= 0

E0L /wL\ -1/6 t~ 2 wL

E~0.92() sin 0 exp~ sin3 0 ~. (12.12)

x \cI I 3c J

The field

iwLV116 I 2wLE~= 0.92E0 () sin 0exp~ si n

3 0 (12.13)t3c J

is called the pump field. Thus in the resonance

E=E~/~=LE~/x. (12.14)


26/102

1 98 IL. Bobin, High intensity laserplas ma interaction

E0 is better written as

E0= F0 1/2~(T) (12.15)(2irwL/c)

where ~ is the function

~ Texp(~r3) (12.16)

of the variable

r (wL/c)113 sin 0. (12.17)

Equation (12.14) was written in the absence of damping. It exhibits an unphysical divergence at the

origin. Now, denoting by i. an effective collision frequency which processes others than actual

electron ion contribute to,

(12.18)

w(w lv)

with the consequence

E(nc)=~Ep, (12.19)

a finite value. The absorbed intensity in this situation (the so-called resonantabsorption) is

0

r E21=2 I -dx. (12.20)J 8ir~~

In a linear profile

~ (12.21)

LL \ L/wi L2 \ L1w2

so that using (12.15) together with a constant v

E2 ___

2 (12.22)

Now, as shown on fig. 12.4, /(r) has a maximum close to 1 when r is in the vicinity of 0.7. Then the

fractional absorption I/Ia, which is independent from v, may be as large as 50% provided the angle ofincidence, which determines r , is suitably chosen. Measurements of the electron temperature [45] or of


27/102

IL. B o b in , High intensity laser plasma interaction 19 9

c:~(~t) Te Experimenis

1 IA P Polcirizahon

0.7T

Fig. 12.4. Plot ofb(r) versus r. Fig. 12.5. Results of T~or I / Jo measurements versus 0 , for bothpolarizations.

the fractional absorption [46,47] clearly e vidence the effect of the resonance. The general behaviour ofthe experimental results is sketched on fig. 12.5.

13. Electron bunching Cavities

The ponderomotive force term (11.5) applies to both electromagnetic (laser) and longitudinal

electron plasma waves. The force acts directly on the electrons. Thanks to the charge separations field

which pulls back the electrons, it eventually acts on the plasma as a whole. One, of course, has to

distinguish between high frequency phenomena in which ions may be considered as a motionless

neutralizing background and low frequency processes in which both species move together. In the two

cases the electron density profile has to be calculated consistently with the field, the ponderomotive

force providing the coupling.Let us first look at the standing wave pattern found in the previous section. It results from an

electromagnetic wave impinging onto a linear density gradient. Electrons feel the ponderomotive force

which pushes them towards the nodes as shown on fig. 13.1. Obviously, the linear density profile is

inconsistent with the standing wave. In order to investigate what is to happen, consider first thegas-dynamical equations

(13.1)

3u 0U \ n ~E2 9n (13.2)\3t ax! n~8irax 3x

EtNn~ perturb~ n ~

Fig. 13.1. Electron bunching towards the nodes of a standing elec- Fig . 13.2. Evolution of an electron density perturbation near cut-off.

tromagnetic wave. Cavity formation.


28/102

2 (X ) J.L. Bobin, High intensity laserplasma interaction

in which ~udenotes either the electron or the ion mass. Moreover the averaging bracket () has been

definitely dropped. The ponderomotive forces in the right-hand side of (13.2) involves a field which is

given by the normal incidence standing wave equation

~2 a2 E /ax2 + w2(1 n/n~)E= 0. (13.3)

(13.1) to (13.3) are a complete set of coupled equations. It can be linearized around a density profile

N(x) and a zero initial velocity. One thus gets for the density perturbation

a2n 1 iaE2aN a2E~ a2N(13.4)

a t 2 8lTflc \ ax ax a x 2 / a x 2

Any perturbation is to grow up. To be more specific let N(x) be linear. Then a 2 N / a x 2 is zero. At cut-off8E2/ax is negative and 82E2/8x2 is zero so that a2n/at2 is negative. At points where aE2/ax is zero, i.e.the extrema of F2, a2n/a t2 have the sign of a2E2/ax2: negative for the maxima, positive for the minima.Hence cavities build up in the electron density profile specially in the vicinity of the critical density.

Since the density at the closest mode may grow up to overdense values, the cavity may then trapradiation.In order to investigate transient behaviours, one has to use a time dependent equation for the field.

This can be done by setting

E= E(t) ei~0t (13.5)

in which the amplitude E(t) is a slowly varying function of time, whose second derivative can be

neglected. One then gets instead of (13.3)

2 iw oE/at + a2 32E/ax2 + w2(1 n/n~)E= 0. (13.6)

Here a2 stands either for c2 or for 3v~,electron thermal velocity squared, so that transverse andlongitudinal waves are simultaneously dealt with; eq. (13.6) is a Schrodinger equation. Since it iscoupled to (13.1), (13.2) the whole system is non-linear. The behaviour of certain classes of solutions of(13.6) can be evidenced by playing again with a factorization of the (13.5) type, let then

E = A(x, t) exp{iO(x, t)}; (13.7)

substituting into (13.6), the equation splits into a real part

2~A+ ~ (0~)2A+ (i A = 0 (13.8)

and an imaginary part

a2 a2A0+O~A~+0~~A=0. (13.9)

w 2w


29/102

.J.L. B o b in , High in ten si ty la serplasma inte ra ct io n 2 01

Introducing a hybrid velocity

2

u=~O~, (13.10)w

assuming A has a slow x dependence: ~ 0 , and linearizing again around N(x), one gets

Du u 3u 1 aN=+u= (13.11)Dt at ax 2N~ax

(13.12)

Equation (13.11) is similar to the Lagrangian equation of motion of a particle in a gravitational field.The corresponding particle is decelerated when moving towards higher densities and accelerated in theopposite direction. According to (13.12), the amplitude A of the electric field perturbationwith velocityu increases logarithmically al l the way down the density gradient.

Such results were obtained by Morales and Lee [48]using numerical simulation. The electron densityprofile is initially linear. The longitudinal field results from a capacitor model. When ions are notallowed to move, dips in electron density associated with soliton-like pulses of electric field, are

generated at cut-offand propagate towards lower densities.Recent investigations [49]about this generation of solitons show for given values of the pump field a

periodic birth of equal amplitude solutions at resonance. By decreasing the pump a double period showsup. A further decrease causes some chaos to set up: random amplitudes separated by random timeintervals. The behaviour changes when using movable ions. A cavity begins to form at cut-off and is

progressively transformed into a shelf. This effect was also evidenced by De Groot and Tull [50].It is

illustrated on f ig. 13.3.

/

Fig. 13.3. Unstable density formation: (a ) initial density profile, (b) cavity at cut-off, (c) final state.

14 . Steady structures, profile steepening

In order to investigate which structures dominated by the ponderomotive force may take place, onehas to start from the first integrals (11.9) in which the thermal flux F is neglected, an assumption to be

justif ied a posteriori in section 16 . The radiative pressure is given by (11.3). The electric field of the laserwave satisfies the time independent Schrdinger equation


30/102

20 2 fL. Bobin. High intensity laserplasma interaction

C2 d2E F fix! \ Vci

I+ ~ 1) _-_JE= 0 (14.1)wdx w V

in which absorption is accounted for.

Several steps are needed to determine qualitatively the density profile. First it is not unreasonable to

assume that the field is to have a standing wave structure with zero field points (nodes) as in the linearprofile dealt with in section 12 . Then p is eliminated between the first integrals (11.9), yielding a

quadratic equation in V(y =

4J2~5(J2V+pp)V+J2~2+5PV2~_E ~~=0. (14.2)J 4ir

The knowledge of E(x), PL(X), 1(x) allows one to calculate V(x). The shape of the profile can beinferred by noting that absorption is important only in the vicinity of the critical density. Furthermorethe points such that E(x) = 0 are singularities of the system. They are located in the PLVplane on thehyperbola whose equation is

4J2V2 5(J2V0+p~~pL)V+ J

2V~--5Pi:uV~= 0. (14.3)

This is the locus of singularities which appear as cusp points of the curve pL(V). They accumulate when

approaching the intersection of the curve (14.3) with the sonic line

8J2V+5(pLJ2V~po)=0. (14.4)

The starting point PL = 0 , V= V0, being subsonic, so is the whole steady structure. The function pL(V)

is sketched on fig. 14.1. There exists a limiting specific volume, an upper boundary for a subsonic

profile.

A steep gradient corresponds to the first arch of the field starting from state 0. Due to the existence ofVjim, there will be a minimum value for the density. Hence an underdense modulated shelf as shown on

f ig. 14.2 .

Density profile steepening due to ponderomotive pressure is a well-known effect which w as widely

observed in both numerical (Valeo and Kruer [51]; de Groot and Tull [501;Forslund et al. [52]) and

laboratory (Azechi et al. [53]; Atwood et al. [54])experiments. Some computer simulations also exhibit

P

~ Vlpm~Fig. 14.1. Ponderomotive pressure versus specific volume. Fig. 14.2. Density and field profiles.


31/102

IL. B o b in , High intensity laserpla sma interaction 20 3

a shelf. Since they have the periodicity of E2, the modulations have a wavelength equal to half that of

the incoming radiation. The maximum length of the shelf is expected to match the coherence length.However high impinging intensities are usually associated with short pulses. One has then to check that

the structure can be built up in a time smaller than the laser pulse duration. In order to do so, one needsfurther information specific to the problem at hand: e.g. mass and degree of ionization of the ions in theplasma. Whenever an underdense structure is evidenced through interferometric diagnostics

(Fedosejevs et al. [55]; Raven and Willi [56]) it extends over a small number of wavelengths, thusindicating an ion sound velocity

C s i = (ZTeIM)~~ (14.5)

not greater than 10~cm sec1 which might be due to a lagging ionization (small Zin the formula) and/ortwo-dimensional expansion.

In the above quoted experiments, light absorption is in the range 20 40% . The discussion in section

10 emphasizes the role of re-radiation to lower the useful flux, i.e. that part of the incident flux which

actually drives the flow. Another cause for this effect is the creation of superthermal electrons withmean free paths much longer than the scale length of the structure. Part of them are a mere loss to beadded to 4~in the equations. Since superthermal electrons are the result of non-linear effects,

re-radiation, by increasing the ratio E2/n~T,also enhances their production. Thus in the experiments theuseful absorbed flux is indeed a small fraction (


32/102

204 fL. Bo bin , High intensity laserplasma interaction

pressure. An opposite case of strong absorption is depicted in fig. 14.4. It corresponds to a single peak.

This is an unrealistic situation for an electromagnetic field. However, eq. (14.6) also holds forlongitudinal electron thermal velocity squared. It is then clear that fig. 14.4 may correspond to the

resonant peak associated with an obliquely incident p-polarized wave. The corresponding density andlongitudinal field are displayed in the figure. Finally, if only the first part of the trajectory, down to the

first well, is taken into account, it can be identified with the leading edge of a subsequently uniformfield: see fig. 14.5. One then recovers the case dealt with in the absence of non-linearities: absorption atcut-off (Fauquignon and Fbux [6]) with a skin-effect structure.

So far, density and pressure profiles have been discussed for a single-component fluid. Thecorresponding temperature profile is then readily obtained. In the subsonic low-density shelf, thetemperature is high and undergoes only small amplitude variations. It can be considered as uniform.

Now, owing to the strong density gradient in the structure, ion heating through the relaxation with

electrons is expected to be ineffective, in contrast to the case of heat transport dominated flows. There it

was shown (see section 7) that Te/TI is, at most, of order 1.5. As confirmed by numerical simulations

(Forslund et al. [49]), it is not unreasonable to assume that in the shelf of a ponderomotive pressure

dominated flow, the electron temperature greatly e xceeds the ion temperature, i.e.

~Te~T. (14.7)

Consequently, the sonic point is such that the velocity is the ion sound velocity (14.5). Hence the

subsequent rarefaction, ifisothermal, has an exponential density profile,

p = Pc exp(xlc1t). (14.8

E2 p

TFig. 14.4. Resonance absorption: (a) potential w ell, (b) density and Fig. 14.5. Strong absorption in the overdense plasma: (a ) potential

(E

2) profiles. well, (b) density and (F2) profiles.


33/102

IL. Bobin, High intensity laserplasma interaction 20 5

The comparison with the high energy part of experimentally observed ion spectra (see, for example,Campbell et al. [57]) indicates guiding electron temperatures of the order of several keV. However,actual spectra have astructure more complicated than the simple law (14.8). They have been dealt within recent numerical (Shvarts et al. [58]) and analytical (Mora and Pellat [59])investigations.

The case of the shelf proper is also of interest. When setting up an equation of motion, the electrical

neutrality of the plasma was assumed to hold everywhere in the flow

Zn~(x)= ne(x). (14.9)

The actual link between ions and electrons is the electrostatic potential . which obeys a Poissonsequation

= 4[l(~e Zn 1). (14.10)

Since both sides of (11.8) vanish, the electrons are Boltzmannian

fleflee~ (14.11)

with the reduced potential

i/i= = e co s / T e E2IfleTe (14.12)

andn ~is areference density. Now, V~appears explicitly in (2.12)w hich for a plane one-dimensionalflow is

readily integrated to give

J2(P S = ZeM

1n~ (14.13)

Since ions only carry inertia, the density modulations in the shelf are modulations of n 1 . Hence a more

deeply modulated ~ pattern. Then from (14.12) one deduces c l i and the corresponding electron density

profile which exhibits, in the nodes ofE, the expected bunching due to the ponderomotive force: fig.

14.6.

X

a

Fig. 14.6. Electron and ion density profiles (a) and potential wells (b) in a ponderomotive pressure dominated standing wave pattern.


34/102

20 6 IL. Bobin, High intensity laserplasma interaction

The situation thus involves BGK-type equilibria [60]with cold ions and electron trapping. There is

some analogy with what happens in the structure of collisionless shock waves (Montgomery and Joyce[61]). However, in the present case, one has to deal with multiple potential wells which is more

complicated.

15 . Isothermal regimes

Some other propertiesof laser driven steady flows are determined after a very simple model in which

an isothermal plasma is assumed. In the one-dimensional plane case, the first integrals reduce to

pu=J (15.1)

p+PLF/p=po+J2/pt). (15.2)

This system can be solved for PL as a function of V. i.e.

PL= J2VpoVo/V+po+J2V0. (15.3)

On fig. 15.1 are shown several special cases selected for different values of the mass flux J . The

maximum value of PL decreases from p~to zero as J increases from zero to Po V~(sonic conditions in

state 0).When the mass flux J is zero, PL is a monotonic function of V. It increases from zero (V= V0) and

tends asymptotically towards p0(V~x). A round trip starting from the point (V0, 0) along the curve,

describes a cavity inside the plasma. The cavity is filled with resonant high frequency electromagnetic field.The tuning depends on the cavity size. Ofcourse the cavity maybe generated by the mechanism described in

section 1 3 (fig. 13.2).

When a plasma flow is present (J0), V(E2) is bivalued. So is partly V(pL). A cavity is compatible

only with the smaller amplitudes of PL. Then V(p~)remains single valued. Beyond a critical Jdependent value of PL, the only allowed profile is the one consisting of a steep gradient followed by an

underdense modulated shelf. Such profiles w ere calculated both analytically and numerically in [62].

The transition from the cavity to the shelf (fig. 13.3) was observed in numerical simulations.

2E p

Fig. 1 5 . 1 . Field energy density and ponderomotive pressure versus the specific volume.


35/102


In the isothermal regime, first integrals can be found also for the one-dimensional spherical

geometry. They read

?pu = J (15.4)

2 ~2 2

U p L~ U0+c~In+-=. (15.5)

2 Po 8 T T p 2

The spatial dependence of the density p is then readily obtained. For a given value of E2, r(p/po) is

bivalued: a subsonic and a supersonic branch are found. A discontinuity in the flow separates aradiation free region (F2 = 0) from a region with a given value of E2 . It can be a subsonic to a

supersonic transition: fig. 15.2, or a supersonic one: fig. 15.3. Both were investigated analytically andnumerically. The subsonic density hump has been evidenced in simulations [62].The underdense part

has a modulated profile. Indeed standing waves may set up. The detailed structure can be found in [63]for the subsonic to supersonic case and in [64]forthe supersonic one. In both, there is no special reasonforthe point at critical density to be sonic.

subsoni ~ (F/P(O) -~-- -- PiP(O) ~

-- ~supersoni~~~~rsupersonic

Fig. 15.2. Subsonic to supersonic transition. Fig. 15.3. Supersonic to supersonic transition.

16. Anomalous transport

In the linear regime, the overdense region is dominated by the thermal transport flux density (7.1).

This term was neglected in the investigation of ponderomotive force dominated structures.

These occur at high laser irradiance. They are associated with low absorption and still with highelectron temperatures, for which the standard theory of heat transport in plasmas [25,26] would predictan increase of the thermal flux. Now there exist experimental evidences for an inhibited transport inlaser interaction with solid surfaces. For instance, Pearlman and Anthes [65] measured the velocity ofions recorded on both sides of a laser irradiated thin foil as a function of the incident intensity: fig. 16.1.

Beyond a threshold of 7 x 1 01 3 W/cm2 most of the energy goes to the plasma on the laser side thusdemonstrating transport inhibition.

Furthermore, Lagrangian computer codes devised for laser driven implosions, do not restitute theexperimental results for high intensity irradiance, whenever (7.1) is used. The agreement is much betterif a significantly lower flux is introduced in the calculations. This is usually done phenomenologically by

applying aflux reduction factor to the free streaming limit


36/102

20 8 IL. Bobin. High intensity laserplasma interaction

F o ilOfl ~ baser

velocity R

id3 iO~~\N.crrj~

Fig. 16.1. Ion velocities on both sides of a laser irradiated foil.

nT T 1/2FL=fFFS=f(-) . (16.1)

4m

FFS is the maximum energy which can be transported in one direction by a density n of electrons with

temperatureT

Using (16.1) guarantees that no energy greater than that available in the Lagrangian cellis carried away owing to large gradients. A hybrid transport term such that

1 (16.2)

F xT52(aT/ax) FL

is the simplest way of accounting for limited transport in numerical codes.

It turns out that the best agreement between experiment and simulation is obtained when in (16.1),f

has the magic value

fm0.03. (16.3)

The limited free streaming flux FL may also be expressed in terms of gas dynamical variables, i.e.

F L = hpc~ (16.4)

with a limiting factor

h ~ (165 4(Z+ 1)3/2 ~m)

To fm corresponds a magic value of h of order 1 , so that FL is also

1/2

FL~nT(M ) =nTc5. (16.6)

Thus, electrons in the transport process carry their energy Twith at most the ion acoustic velocity. Ionacoustic waves do not involve charge separation. The semi-empirical formula (16.6) may then tell us that


37/102

IL. Bobin, High intensity laserplasma in ter a c tio n 209

no charge separation takes place in electron thermal transport. This could be a guiding idea for

investigations of the flux limited transport problem. A further discussion of the point is to be given inthe next section. For the moment, it suffices to observe that no definitely convincing theory is available

yet.

Here, it will be simply assumed that formulas such as (16.1), (16.4) hold. Then, following Cowie and

McKee [66], consequences on a steady gas dynamical regime will be examined. First the energyequation (7.2) is rewritten

V~[pu(~u2+ ~c~)+hpc~c~]= 0. (16.7)

Hence the first integral

~Y?(~P?2+5)+h=0 (16.8)

in which, as in section 7, the integration constant is set equal to zero. It results from (16.8) that the

Mach number is a constant across the whole structure. This w as first pointed out by Cowie and McKee

in an astrophysical context.

In the plane one-dimensional case, the first integral of the equation of motion associated with (16.8)only yields the trivial solution: all quantities are constants. Or alternatively, there exists a discontinuityin the flow [67]. It separates a subsonic state from a sonic state. When the ponderomotive force is

accounted for, its structure is the one investigated in section 1 4.For the spherical case, using p = pc~,(7.10) becomes

~Pl2pr2=JU, (16.9)

yielding

p/pa= (rIro)_2m2~n21). (16.10)

By the same token, after (7.23)

T/T0 =(r/ro)4/s1), ~ = (T/T0)2(m

22)1(m?l). (16.11)

Such profiles have slopes much steeper than those found in section 7 , assuming a so-called classical

transport.

The flux limited transport term is used in gas dynamical equations. It was assumed in section 14 that

FL is much smaller than the absorbed light intensity I . Hence the condition

nT T~2fi

( ; ; -)


38/102


3 i 3/21= ~ a . (16.13)

2(yl) p

Then, the inequality (16.13) is rewritten

p32 (16.14)

4 m p

Accordingly a condition forfis obtained

f< 12(m/M)2. (16.15)

It does not depend on the laser wavelength. Obviously the magic flux limiting factor fm fulfills the

condition. Thus, the assumption made in section 1 4 is justified. Physically, a dominant ponderomotive

force and a strongly inhibited transport apply together in laser plasma interaction. The influence of the

thermal transport is a small correction which does not reduce the steep gradients in the vicinity of the

critical density.

17 . Possible physical origins of flux limitation

It is of course a rather unsatisfactory situation, that an ad hoc flux limiter be introduced withoutreferring to any physical process. Accordingly, attempts have been made towards an understanding of

the physics of heat transport in laser irradiated plasmas. The problem can be stated as follows: given a

region in a density gradient, how does heat propagate towards higher density zones? Furthermore thedriving electron distribution normally has two temperatures and the regime may be unsteady. A first

question arises: which physical processes should be accounted for?In a thermal conductivity such as (7.1), only electron collisions with charged particles are considered.

The usual theory is steady state and one dimensional. In the plasma kinetic theoretical calculations by

Spitzer and Harm [25] or Braginski [26], it is assumed that the gradient scale length is larger than themean free path. A local theory then applies. Now, Spitzer and Harm derivation of transport coefficients

do include flux limitation. Indeed thermoelectric effects should be also dealt with, since the electrons

which participate in the heat transport generate an electric current. One then has to solve coupled

phenomenological equations for the current density j and the heat flux F, which read with positivecoefficients

juEaVT (17.1)

F=f3EKVT (17.2)

in which, of course due to the laws of irreversible thermodynamics, the coefficients are not independent.The main point here is that for steady state, j should be zero, otherwise electrical neutrality would nolonger prevail and the electric field would increase without limits. Setting j = 0 in (17.1) yields a local

electric field. The heat flux is accordingly


39/102

IL. Bobin, High intensity laserpla sma interaction 2 1 1

F = K(l af3IuK)VT (17.3)

in which a reduction factor due to thermoelectric effects shows up.

However, this is still too large to be consistent with experimental results on laser driven plasma

dynamics. Then, one may argue that mechanisms not accounted for in collisional theory should be

investigated and their effects on heat transport evaluated.First of all, as usual in plasma physics one might think about instabilities. The heat transport problem

may be stated as : given any heat flow, it can be proved after the theory of moments (up to the 14th) thatone can always find a distribution function which carries the specified flux [68]. Such a distribution

includes a beam and is strongly unstable. The maximum allowed heat flow in the problem at hand is

then the one for which the electron distribution is still stable.

Since as shown in (16.6) there seems to be some connection between thermal flux reduction and ionsound, it is no surprise that the onset of ion acoustic turbulence w as the instability most widely

investigated. Long ago, Fried and Gould [69] showed that an electron drift is able to trigger an ion

acoustic instability. They calculated a critical drift velocity as a function of the ratio Tell1. Whenever

the drift velocity is greater than critical, the instability sets in and the energy goes to turbulence instead

of being transported.

The way one-dimensional electron transport results into a drift is illustrated on fig. 17.1. With respectto equilibrium, the electron distribution function has a larger tail in the direction of the energy flow. In

order to ensure a zero net current, the bulk of the skewed distribution function is displaced towards the

opposite direction. The homogeneous growth rates of the instability were calculated by kinetic theoryboth for collisional [70, 71] an d non-collisional [72, 73] situations. They coincide in the limit Te ~ T 1 .

Now, the non-collisional growth rate turns out to be density dependent [74]. In a ponderomotive force

dominated laser driven plasma flow, the growth rate is higher in the vicinity of the critical density where

steep gradients prevail and are likely to inhibit the instability. For the underdense shelf, the growth rate

is much smaller, so that in this case also the instability does not set up easily. However there has beenexperimental evidence of ion acoustic turbulence in plasmas created with comparatively low laser

intensities [75,76] .The ion acoustic turbulence also acts directly on the transport coefficient through an effective

collision frequency. The point is far from being clarified yet and large discrepancies exist between

theoretical evaluations (see [77]).

So the role of electron heat flux driven ion acoustic turbulence, in reducing transport is not firmly

evidenced.Another source of flux limitation is also found in a kind of electromagnetic turbulence involving

We ibel microinstabilities. Indeed, since the temperature gradient causes an anisotropy in the electron

r(v)

AcFual

~~Mawellian

O~vEnergy Flow

Fig. 1 7 . 1 . Heat carrying electron distribution compared to Maxwellian.


40/102

21 2 IL. Bobin, High intensity laser plasma interaction

distribution function, low frequency electromagnetic modes become unstable [781. The chaotic magnetic

field thus generated was shown to effectively reduce the electron mean free path [79]. Saturation isexpected from mode coupling. At the saturation level, the electron mean free path turns out to be

consistently smaller than the temperature gradient scale length L. The limitation factor in (16 .1) isaccordingly

f-~(c/Lw~)~ (17.4)

with the effective collision frequency

Ve LW 1/3Peff~~(~) (17.5)

which appears to be larger than the values calculated for the ion acoustic turbulence.

One more candidate for transport inhibition is the self generated D.C. macroscopic magnetic field

(see section 20). Across such a field whose order of magnitude can be megagauss, heat transport is

mainly due to ions whose diffusion coefficient is expected to be somewhere between classical (oB2)

and Bohm-like (ceB1). The influence of magnetic fields on electron transport was recently demon-strated [80] in an experiment involving several laser beams impinging onto different locations regularlyspaced on the surface of a rod-like target. X-ray emission from the hot plasmas thus created, allows

pinhole imaging. Hot spots are naturally observed at the laser foci. Some others also appear at discrete

intermediate locations corresponding to zero field point in the magnetic field pattern resulting fromlaser interaction in such a geometry. Energetic electrons emitted by the plasmas concentrate again in

such points as shown on fig. 17.2. The magnetic field both contributes to the inhibition of heat transport

perpendicularly to the lines of force and favours energy transport to definite places by superthermalelectrons. Details on the influence of D.C. magnetic fields on electron transport are given in the review

by C. Max [77].In all the above investigations, the plasma is considered classical and dilute. This might not be always

the case in laser interaction. When a frequency converted Nd-laser beam impinges into silica or usualmetal targets, the resulting temperatures ( 0 . 5 1 keV) at critical density correspond to an average ioncharge number (Z) of about 10 . Thus the ion plasma parameter

F, = (Z)2 e2(ne)3/T 1 (17.6)

t .~

O~\ / ) ~ )

Fig. 1 7 .2 . Electron trajectories (~) in presence of magnetic fields (0 ) generated by laserplasma interaction.


41/102

IL. Bobin, High intensity laserpla sma interaction 21 3

T(kev)

1cu t oF f ~

1 0 ~ incr~osingdecreasing Z ~

-

Id schocked region

fl~cm

10 1 02 3 1 0 2 5Fig. 17.3. Temperature/density profile in the conduction zone of a 0.25 ~smlaser irradiated high Zmaterial. The effective ion charge may decrease

when going from the critical layerto the ablation front. However, the ion plasma parameter is still high.

a value which is conserved all along the profile on the high density side (fig. 17.3). Consequently heat

transport takes place in a strongly correlatedmedium, the potential energy between neighbouring ionsbeing of the same order as the kinetic energy. The influence of strong correlations on ion thermalconductivity was studied numerically by molecular dynamics in a one component plasma, i.e. in whichthe electrons are a neutralizing fluid [81]. Of course, as before, the electric current is zero in the

one-dimensional problem. It turns out that for a F of 0.5 or 1 the transport coefficient is significantlyreduced (roughly by a factor of 2) with respect to the usual values for the same densities and

temperatures.

Transport in the region of laser driven plasma flows might also be due to the spreading of thermalradiation. Indeed radiative phenomena are expected to be important for high Z target irradiated by

high frequency lasers. Soft X-ray emission is widely used as a diagnostic [82]. Reportedly [83]indirect

implosion schemes use thermal radiation. The corresponding transport problems are scarcely dealt with

in the open literature. However some results are available. For instance it was shown in [84] that lateralradiative transport in layered targets induces a pressure smoothing. Moreover any important transverse

energy transport contributes to flux reduction in the direction of the laser beam axis.

18 . Flux limitation from transport theory

The usual SpitzerHrm derivation of transport coefficients includes a flux limiter thanks to

thermoelectric effects. However, assumptions were made which do not apply to laser driven plasmaflows: thick (with respect to the electron mean free path) conducting zone, gentle gradients and small

distortions of the equilibrium (Maxwellian) electron distribution function. Extensive research has been

recently carried on transport theory.The starting point is the kinetic equation for electrons

(18.1)a~ a~- 3v t~

in which t9f/otI~is any realistic convenient collision term.


42/102

21 4 fL. Bobin, High intensity laser plasma interaction

A standard technique for the numerical solution of this equation uses a splitting of the distributionfunction into an isotropic part f(0) (Maxwellian) and an anisotropic part

(18.2)

where ~t = vjv is a small parameter. The heat flow can be calculated by integrating v3~1~over theentire velocity space

J v3f~(v)d3v = 4ir J v5J~(v)dv. (18.3)

It was shown by Gray and Kilkenny [85]that in the SpitzerHarm theory, the heat is mainly transported

by electrons in the high energy tail (fig. 18.1) and the total electron distribution function is not positive

for all ,a. This is unacceptable. In order to avoid it, the anisotropic part of the distribution function has

to be smaller than the isotropic one, i.e.

(18.4)

for those velocities which correspond to an important heat flow. The condition implies a smaller flux.

However, using a spherical harmonic expansion and considering f (O), f(l) as the moments of thedistribution function, one gets [86] fluid dynamical-like equations

(~+~ v)t0 a[g _~O)]t9 t (18.5)

( - ~ -+vp1~v)f~=-~{V1 vJ~f~-f~f~}+agf~}

in which the indices refer to the components of f(l) ff2); g (>0) is a given source, and a is a relaxationconstant. Then if the initial condition is Lf 11 < 1 everywhere, it cannot be subsequently greater than 1.

Indeed in the r.h.s. of the second equation (18.5), the last term acts to decrease ~ in magnitude

whereas when Lf~11I* 1 the first term vanishes as 1 {f(l)J2 The equations are automatically flux limiting.Furthermore, they have the same free streaming solutions as the exact transport equations in the limitof a very thin plasma layer.

Many attempts have recently been made in order to get quantitative results and a large com-

L F~

~) ~V F1

Fig. 1 8 .1 . Spitzer Hrm values for electron energy distribution, heat flow and the ratiof, /f o. L is the gradient scalelength. A the electron mean freepath.


43/102

IL.Bo bin , H igh intensity laserplasma interaction 21 5

putational effort is going on [87].Most of it is grounded on a kinetic theoretical approach, the collision

term in (18.1) being of the FokkerPlanck type. Usually one uses a Legendre polynomial expansion:

f(x, v, ~i, t)= ~ f~(x,v, t)P~(~) (18.6)

with a truncation at some specified fN~The resulting set of coupled equations for thef~is thenintegrated numerically with a zero current constraint and given initial and boundary conditions,

examples of which are displayed on fig. 18.2. The transport is one dimensional but of course thedistribution functions are 3-D.

Initial conditions with a temperature step inside a uniform density plasma slab were used by Bellet al. [88]. Their computation shows an evolution towards a temperature gradient which extends overthe whole slab. The heat flux is plotted at each grid point for a given time as a function of the local ratioof the gradient scale length to the electron mean free path LIA. The maximum heat flux is always found

for LIlt


44/102

21 6 IL. Bobin, High intensity laserplas ma interaction

diffusion procedure for those electrons whose velocity exceeds some prescribed value v (-~2ve). Thetemperature profile is then qualitatively different (fig. 18.4), and very likely is more realistic. Indeedalthough experiments are seldom clear, it can be inferred that comparable energy fluxes are transportedwith mean free paths which differ by an order of magnitude. In laser irradiation of thin foils, two-sided

measurements such as those at Rochester [91] indicate a stronger flux limitation than when measure-ments are made on the laser side only [92].Now, according to D. Shvarts, in a temperature profile suchas the case b) of fig. 18.4, half of the energy leaving the interaction region is deposited in the steep

gradient and contributes to imparting momentum to the target. The other half is deposited far beyondpoint A. Its role is mainly preheating. Measurements on the laser side indicate which (limited) flux isleaving the interaction region whereas on the opposite side many properties depend on the (twice asmuch limited) flux deposited in the steep gradient zone.

The next step in the theory deals with strongly inhomogeneous plasmas. A lot of computational work

is presently being done [87], little of which is so far published.

An alternative numerical approach to heat transport across steep density gradients uses particle

simulation or rather a mixture of particle and fluid dynamics [93]. A Monte-Carlo method is used topredict the behaviour of the particles. In such a simulation, coronal decoupling of the electrons in thedensity gradient and the subsequent electric field which drive the return currents are automatically

accounted for. The comparison with flux limited calculation is better made for the temperature profile,f ig. 18.5. There is no value of the flux limitation factor to match the results of the Monte-Carlo

simulations over the whole heat conduction region.

The thermal flux at some given point inside a thin slab depends upon the distribution function at

different places throughout the temperature gradient. This non-local behaviour is automatically ac-counted for in the above numerical computations. On the contrary, in the Lagrangian fluid dynamical

codes extensively used to describe laser plasma interaction, the thermal flux is modeled by means of a

local formula of the (16.1), (16.2) type. In order to reconcile the actual non-local character of the heat

flux with a simple formulation acceptable by fluid codes, Luciani et a] . [94] proposed the followingexpression

F(x) = J K(x, x) FSH(x) dx. (18.8)

Te

Te

f Ui ~

//Flux limi~edlranspor~

_ _ _ _ _ _ _ X Monte Carlo calculationsFig. 18.4. Temperature profiles obtained computationally: (a) usual Fig. 1 8 .5. Comparison of Monte-Carlo calculations after Mason [93]

f l ux limited transport, (b) non-local hybrid model, with temperature profile resulting from transport theory with different

f l ux limiting factors.


45/102

IL. Bobin, High intensity laserplasma interaction 21 7

FSH is the SpitzerHarm flux, and K is a temperature dependent kernel

1 / ~ n(x) dx \K(x, x)=

2A( ~)exP~ ~ A (X1 ) n(x))~ (18.9)

A is an effective electron range

aT24irn(Z+ 1)12e4lnA (18.10)

For gentle temperature and density gradients, the kernel K(x, x) reduces to a 8 function and onegets the usual Spitzer.-Harm value. On the contrary, in the case of a step-like temperature gradientassociated with a uniform density, the exponential in (18.9) can be approximated by 1 at the location of

the temperature jump. A straightforward integration of (18.8) yields a maximum heat flux which is afraction of the free steaming value, provided the adjustable constant a is conveniently chosen: a value

of 32 restitutes the results of FokkerPlanck simulations [89]. Furthermore the model predicts aprecursor and except for this particular feature is in agreement with a flux limitation factorfof

about 0.1.

19 . Filamentation and self focusing

So far, mainly one-dimensional situations were considered. We have seen how in high intensity

interaction, the ponderomotive force acting in the direction of light propagation at normal incidenceresults in density profile modifications. The wave nature of electromagnetic radiation is essential. The

structure of the plasma flow matches the standing wave pattern due to the reflexion at cut-off. Throughcoupling via the ponderomotive force, E.M. waves and the plasma flow behave consistently with each

other.Of course onemay expect similar results in two (or more) dimensional situations. For instance, when

light impinges obliquely into a density gradient, the ponderomotive force acts in both directions parallel

and perpendicular to the gradient. The effect was investigated by means of numerical simulations which

show the building up of a diffracti

high intensity laser plasma interaction.pdf

Documents