fulltext

Study of hydrodynamics of laser plasmas

produced from thin foil targets (*)

L. FERRARIO and D. BATANI

Dipartimento di Fisica, Università di Milano - Via Celoria 16, 20133 Milano, Italy INFM, Unità di Milano - Milano, Italy

(ricevuto il 21 Marzo 1996; revisionato il 27 Agosto 1996; approvato il 30 Settembre 1996)

Summary. — The hydrodynamic expansion of plasmas produced by laser beams

focused on thin foil targets is studied by using a self-similar approach. This work constitutes a generalisation of London and Rosen’s model proposed in the context of X-ray laser studies. Here two-dimensional cases and cases with non–flat-top laser pulses are considered, for which analytical scaling laws are deduced. The model is useful to study long-scalelength plasmas, produced by uniform laser irradiation, which become fully underdense during the laser pulse. It predicts the fundamental plasma variables (scalelengths, density, temperature) as a function of the experimen-tal parameters (e.g., laser intensity, wavelength and pulse duration, target thickness and composition).

PACS 52.50.Jm – Plasma production and heating by laser beams. PACS 52.40 – Plasma interactions.

Introduction

The development of easy and quick methods for predicting the dynamical behaviour of plasma expansion holds remarkable interest for the study of laser-produced plasmas, since they can support more expensive and time-consuming computer simulations. In particular, considering plasmas produced from thin foil targets, in 1986 London and Rosen [1] proposed a simple one-dimensional self-similar model for applications in the framework of soft–X-ray laser studies [2]. The use of a few assumptions (homoge-neous expansion of the plasma, uniform electron temperature) permits to reduce the partial differential equations of hydrodynamics to a set of coupled ordinary differential equations. Besides, by using suitable temporal shapes of laser pulse (London and Rosen used a flat-top intensity laser profile) it is possible to solve analytically the hydrodynamic equations, finding useful power-law scaling relations for the fundamen-tal conditions in the foil plasma (e.g., scalelength, density, temperature).

Nevertheless, London and Rosen’s model presents some severe limits. Indeed, since the model is 1D, analytic and numerical results are realistic only as long as the dynamics does not involve lateral expansion of plasma. This is expected to happen at

(*) The authors of this paper have agreed to not receive the proofs for correction.

L.FERRARIOandD.BATANI

46

times of the order of t2 DB 2 R O cs[3], where R is the laser focal spot radius and cs the

plasma sound velocity. Since in many experimental conditions t2 DE tL (tL being the

laser pulse duration) the evolution of the plasma is expected to be subject to two-dimensional effects. This is especially true for low-energy nanosecond laser systems which need high focalisation to reach the intensities (F1012 WOcm2) needed in most laser-plasma experiments. These are the systems more used in medium-size laboratories. Besides, at early times (when the influence of initial conditions is not yet lost), it is not possible, as described below, to apply a self-similar model, even if the plasma is characterized by a one-dimensional expansion.

Another limit of London and Rosen’s model, in the search for analytic solutions, is represented by the use of flat-top pulses, not a very realistic shape. In fact, experimentally, in most of the cases of interest, the laser intensities have a Gaussian temporal profile.

Even if indeed some experimental works have shown the possibility of working with 1D expansion and flat-top laser pulse, this requires lasers with very high pulse energy and it is not a very common experimental situation [4].

As a generalisation of the one-dimensional London and Rosen model, the 2D problem has been considered by Hunter and London [5] with particular reference to the search of analytic solutions for constant and laser heating and for adiabatic and isothermal expansion. In the «constant» heating case the plasma is still dense enough to give a complete absorption of laser light. Hence it produces a constant heating in the case of flat-top laser pulses. When the plasma becomes less dense we speak of «laser heating». In their work Hunter and London use Cartesian coordinates. Anyway, for comparison with many experimental situations, it is useful to adopt cylindrical coordinates in the solution of the hydrodynamics equations. This requires one equation less than Hunter and London’s model, so that it is possible to obtain a simpler, and closer to experiments, model.

After having reduced the hydrodynamics system, by means of similarity hypothesis, to a set of ordinary differential equations in cylindrical coordinates (sect. 1), we compare, in the following section, the 1D analytic and numerical solutions, by using a «nearly-trapezoidal» (n.t.) laser pulse. By imposing appropriate «resemblance» conditions, the solutions for n.t. pulses may be used to simulate a Gaussian shape (sect. 3). In fact, in the last case, it is very difficult to find scaling laws in all time periods during plasma’s expansion (sect. 4). Finally, analytic solutions of similarity equations in cylindrical geometry for special cases are presented in sect. 5. Some more details may be found in [6].

1. – The similarity equations in cylindrical coordinates We start with the usual ideal hydrodynamic equations:

.

`

/

`

´

g

h

g

h

k

g

h

l

where r, p, v are the mass density, pressure and velocity, e and H are the internal energy and the net heating rate per unit mass, respectively.

The system is solved assuming a cylindrical symmetry around the axis representing the beam’s propagation, which is assumed to be perpendicular to the target surface. Similarity equations will be obtained assuming that plasma flows are isothermal in space and with a velocity profile linear in space:

vi(t) 4 xi Li(t) d Li(t) dt , (2)

where i 4x, r and the longitudinal and transversal coordinates, while Liare the plasma

scale lengths. The momentum equation can then be separated in space and time. We will now assume (as done by London and Rosen) that the plasma follows the equation of state (EOS) and the specific heat per unit mass of a perfect gas, i.e.

p 4r Z * T Mi , cvf de dT 4 3 2 Z * Mi

with Mi the ion mass and Z * the average ion charge. It is then possible to find the

following expression for density profile, normalized with the continuity equation:

r(x , r) 4 M ( 2 p)3 O2LxLr2 exp

y

z

where M 4pr0dRt2 is the total mass of the foil (r0 is the initial mass density of the

foild and d is the target thickness). Besides, for the moment, we consider the laser focal spot filling the whole surface of the foil of radius Rt (see fig. 1).

Such situation is really met in experiments concerning X-ray «spot» spectroscopy and laser-plasma interactions, when it is necessary to create a plasma with sufficiently «flat» profiles in the radial direction [7, 8].

Actually, such a Gaussian profile imposes that the plasma expansion is symmetric with respect to initial position of the target. This assumption will be satisfied only in the case in which the target is symmetrically laser illuminated on both sides. Alternatively, with some degree of approximation, it can be considered to be fulfilled also if we consider sufficiently thin foils, with an appropriate composition and properly irradiated, which «burn through» at very early times. This will contribute to create a plasma which becomes underdense during the laser pulse. Besides, particular experimental parameters (laser pulse duration, target thickness, laser intensity) permit to obtain plasmas with long scalelength and nearly constant temperature, necessary conditions to apply a self-similar approach.

With the EOS and self-similar hypothesis, the hydrodynamic set becomes

Lx d2_L x dt2 4 p r , Lr d2_L r dt2 4 p r , cvr dT dt 4 2p

g

h

where T(t) is the electronic temperature in energy units. In order to find a solution of this system, it is now necessary to perform a calculation of the heating rate H. This term describes the rate at which the plasma absorbs laser light. Also, by looking at energy equation, it is evident that the absorbed energy is fully redistributed over the all plasma volume. Indeed, the plasma temperature is assumed to be a function of time

L.FERRARIOandD.BATANI

48

Fig. 1. – Typical experimental set-up for the production and characterization of long-scalelength expanding plasmas.

but not of space. This allows the real absorption coefficient, which is a function of both spatial coordinates, to be substituted with an average.

For this purpose, first we consider the plasma optical depth along a laser «ray» (characterized by the coordinate r) which takes the expression

t(r) 4



2Q 1Q

K(x , r) d x ,

(5)

where K is the plasma absorption coefficient. In particular, because the plasma becomes underdense during the laser deposition, we can assume that absorption is due only to inverse bremsstrahlung (i.e. collisional absorption). Hence, we ignore reflection of light at the critical surface and other ways of absorption (e.g., resonant absorption or scattering instabilities).

Then we consider the inverse bremsstrahlung absorption coefficient [9]:

Kib4 1 .064*

g

h

where neand ncare the electron and critical density in cm23, l the laser wavelength, me

the electron mass, log L the Coulomb logarithm and Te is measured in energy

units.

Since the variation of log L as a function of ne and Te is very low, we will

London and Rosen [1] (log L 45 is a very common value for laser produced plasmas).

Considering neO ncb1 in the previous formula and integrating along the direction

of propagation of laser pulse x, we find

t(r) 4 k2 6 p3 M2_{Z *}3_e6 log Ll2 c3me3 O2Mi2LxLr4 T23 O 2_exp

y

z

We see that t decreases during the interaction between the target and the laser pulse since both Lx and Lr increase with time.

Finally, the net heating rate, becomes

H(r) 4 I(r , t)

m

[

]

(8)

where I is the total laser flux and m the surface density of the foil, m 4r0d .

Formula (8) points out the different heating due to the laser beam along different «rays» in the plasma. In fact, the absorption of laser energy, according to (8), follows the laser profile along the radial direction.

The intensity of the laser beam coming out of the plasma along the ray r is then

Iu(r) 4I0exp [2t(r) ]

and we can define the average optical depth through the ratio between the power «coming out» and the total input power of the laser (this expression is valid at times large enough for the plasma to become larger than the focal spot radius R; at very early times, the dimension of the plasma is of the order of the target radius Rt which we

assumed to be smaller, and hence the radius R should be substituted by Rt):

exp [2atb] 4



Here I0 (which is still in general a function of time) is the laser beam intensity for a

circular laser beam and the maximum intensity (along the beam axis) for a Gaussian laser beam. This means that, for a Gaussian beam, the laser focal spot radius R is defined through the relation



0 Q

2 pr dr I0exp [2(r2O 2 r02) ] 4I0pR2

(10)

or R B1.2 r0 (where r0 is the radius at k1 Oe of the Gaussian profile). Thus we

obtain atb 42 ln

{



}

L.FERRARIOandD.BATANI

50

Fig. 2. – Time history of atb obtained from eq. (11) (dashed line) and eq. (12) (solid line), respectively. The laser intensity has a Gaussian profile, with Imax4 1 .5 Q 1013 W Ocm2, l 4

1 .064 mm, laser pulse duration tL4 3 ns, laser focal spot width W 4 50 mm, aZb 4 3 .2, aAb 4 5 .7 ,

target thickness d 41.5 mm. The comparison points out the good accordance at all times, in particular for late times.

This expression allows the heating rate H to be determinated through the relation

H 4 I0(t)

m

[

]

Even if this can be integrated numerically, it is more advantageous to look for approximations. In fact, when t(r) c 1, the exponential term in eq. (11) is negligible and the heating rate is H BIOm. We will call this «constant heating» case (heating is indeed constant if I is a flat-top pulse). At later times, when the optical depth is negligible, H BItOm can be evaluated assuming r2

O Lr2b1 («laser heating» case).

Thus, eq. (11) can always be written, in first approximation, as

atb 4 k2 6 p3 M2_{Z *}3_e6 log Ll2 c3_m3 O2 e Mi2LxLr4 T23 O 2_. (12)

Figure 2 shows an example of the comparison between the numerical and approximated evaluation of the optical depth as obtained by using eq. (11) and eq. (12).

time at which atb 41. Then we can assume H B./ ´ I Om , Iatb Om , t b ttrans, t c ttrans. (13)

2. – Analytic solutions for «nearly-trapezoidal» laser pulse

In the search for analytic solutions of the hydrodynamic equations in plane-parallel geometry, we must pay attention to the choice of the temporal shape of laser intensity. Even if London and Rosen’s model deals with flat-top pulses (rectangular temporal profile), a more precise comparison with experimental works forces us to take into consideration different intensity profiles. In fact, in most cases, the laser intensities have a Gaussian temporal profile. Since the hydrodynamic system is not solvable in all time periods by using such a profile, we have been looking at a new intensity profile, which we called «nearly-trapezoidal». Such laser pulse is characterized by intensity slopes given by appropriate powers:

Int(t) 4

.

`

/

`

´

g

h

g

h

where g, d D0, while the times t0and t1represent, respectively, the beginning and the

end of the plateau

(

)

Figure 3 shows the behaviour of intensity for a generic example of n.t. laser pulse. Main parameters are also given.

Reducing eqs. (4) to a 1D hydrodynamic set in plane-parallel geometry (formally

Lr_{K Q), we obtain} Lx d 2_L dt2 4 ZT Mi , cv dT dt 4 H 2 ZT Mi d LxO dt Lx . (15)

Substituting T and its derivative from the momentum equation in the energy equation, we get 1 2 d dt

g

g

h

h

where Lx(t) is the plasma scalelength along the direction of the laser beam. At

a given pulse power and target composition, the transparency time depends on foil thickness. Thus, an increase of its mass will correspond to a higher value of ttrans with a consequent variation of the scaling laws. Changing the target thickness,

the position of ttrans could be in the upward slope of laser intensity, during the

plateau, or in the downward slope. For every case, plasma expansion will be characterized by four distinct time periods, each one described by different scaling

L.FERRARIOandD.BATANI

52

Fig. 3. – Intensity history of a generic n.t. laser pulse. tA and tB represent the times

corresponding to the half-maximum value of intensity I0, while t * divides the plateau into two

equal parts.

laws (we recall that the flat-top intensity model, developed by London and Rosen, was characterized by only three distinct time periods).

Here, we analyse only the more realistic case corresponding to ttransE t0. Actually,

the mass target cannot assume too high values since the plasma must be undercritical for the model to be valid. The work will anyway be similar also for the remaining two cases. In particular, for the case t0E ttransE t1 the scaling laws of the plateau period

reduce to those already found by London and Rosen.

If the foil becomes transparent before the laser intensity reaches its maximum, we can use eq. (16) to find asymptotic power laws for each phase, i.e. we search for solutions of the type L(t) 4L0ta, T(t) 4T0tb. After substituting, we find the plasma

scalelength and temperature:

.

/

´

for t Ettrans, and where

a 4

k

l

By substituting in eq. (6), we find (19) Itrans4

y

k

l

z

After the transparency time, but before the plateau, we get instead the following scaling laws:

.

`

/

`

´

y

z

g

h

y

z

for t c ttrans, where m 43.4Q103 m Imax(Z *9 O2O Mi9 O2) log L l2(e6O c3 me3 O2).

The dynamics of the physical event imposes some limitations on the parameter d (both L(t) and T(t) must increase during this time period), but the old condition

(

from eq. (14)

)

The first period ends at t0, the time corresponding to the beginning of the intensity

Fig. 4. – Comparison between analytical and numerical electronic temperature as obtained from a n.t. laser pulse (dashed and solid lines, respectively). Main parameters are: Imax4 6 .5 Q

1013

W Ocm2

, l 40.53 mm, tL4 0 .45 ns, m 4 3 .6 Q 1025 gcm22. The foil material is selenium. The

choice of a sufficiently high g (in this case we take g 45) provides a good accordance also in the last phase of plasma’s expansion.

L.FERRARIOandD.BATANI

54

plateau. Here we recover the analytic solutions of London and Rosen’s model:

.

/

´

where Cs4kZ * T OMi is the sound speed, while Li and Ttrans are evaluated at t0.

With regard to the downward slope of the intensity, we find scaling laws which are physically not acceptable. Thus, in this time period we assumed the scaling law of an adiabatic expansion, as already done by London and Rosen. Of course this hypothesis, in our case, is more and more realistic as the value of g becomes greater and greater (i.e. the n.t. pulse approaches a step) (see fig. 4).

3. – Comparison with Gaussian pulses

In this section we discuss the hypothesis which permit to compare the evolution of the main plasma variables derived from numerical solutions for a Gaussian pulse with the scaling laws obtained for a n.t. laser pulse. As we will see below, results are satisfactory.

In order to obtain solutions which are of interest for the case of Gaussian laser pulse, we impose some «compatibility hypothesis». This means that the n.t. pulse must be similar to the Gaussian one. In this way the plasma expansion history is more detailed and its laws more similar to the numerical solution of hydrodynamic equations, than by using the simpler flat-top profile introduced by London and Rosen.

1) As usual, we characterize the time duration of a laser pulse at half height of intensity. The first hypothesis consists in the n.t. pulse to have the same duration tL

(FWHM) of the Gaussian pulse. This implies

tL4 21 Ogt12 221 O dt0.

(22)

2) Since a Gaussian laser pulse has a symmetric time history (with respect to the time of maximum intensity t), then we impose the following condition of symmetry:

t * 2tA4 tB2 t *

(23)

which, after a few calculations, becomes

t0

k

g

h

l

3) A necessary condition for the n.t. pulse to «resemble» the Gaussian one, consists in requesting the two to have the same energy:





y

g

h

z

Substituting the last expression into eq. (25), we get tL

o

g

h

4) To describe the same plasma evolution, the two pulses must reach their maximum values at the same time, that is t * 4t. From this we find

tL

o

g

ln I0O IG( 0 )

ln 2

h

where IG( 0 ) is the intensity reached by the Gaussian pulse at the time 0 (where the n.t.

laser pulse begins).

We can thus summarize the four hypotheses:

.

`

`

/

`

`

´

o

g

h

o

g

h

where t0, t1, g, d, ImaxO I0, I0O IG( 0 ) are 6 unknown quantities. Also, because of the

choice of these quantities must be physically justified, we are forced to impose some limitations about their range of validity:

g , d D0 , t0E t1.

(30)

Since the number of unknown quantities is greater than the number of eqs. (29), we decided to put IG( 0 ) OI0B 0 .0072, as done by London and Rosen. This value is low

enough so that it does not influence the evolution of the plasma conditions during the expansion. Besides, we decided to maintain one «free» quantity in order to look for the best similarity between the hydrodynamic solutions for the n.t. and the Gaussian pulse. Indeed, we have chosen g as the free parameter and we have looked for the «best» g through a graphic and analytic comparison. For our test cases, the comparison between the scaling laws derived from analytic solutions for a n.t. pulse with the numerical solutions for the Gaussian one, shows that the best behaviour of the plasma variables are obtained for g B2.9. We get also the other parameter values:

t1O t0B 1 .18 ,

Imax

I0

B 0 .75 , d B9O5 .

(31)

By using these, we can draw the time behaviour of the two intensities (fig. 5). The corresponding scaling law for temperature is represented in fig. 6. They are also compared with the analytic results obtained for a flat-top pulse having the same experimental parameters (in the passage through the different phases, the scaling laws are «connected» to each others by using the last value of one phase as the

L.FERRARIOandD.BATANI

56

Fig. 5. – Intensity history from a n.t. (dashed line) and a Gaussian pulse (solid line). Parameter values, given from eq. (31), are respected.

Fig. 6. – Temperature history obtained for three different laser intensities. The solid and the dotted lines are the analytical laws obtained from a nearly-trapezoidal and a flat-top pulse, while the dashed line is the numerical result for a Gaussian laser pulse. Input parameters are the same of fig. 4, but g 42.9. We distinguish four time periods for the n.t. pulse.

initial condition for the following one). The benefits in making use of a nearly-trapezoidal laser pulse are particularly evident at early times when laser intensity increases with time.

4. – Analytic solutions for Gaussian laser pulse

The direct use of a laser pulse with a Gaussian intensity profile does not permit to obtain analytic solutions from one-dimensional hydrodynamic set for all time periods during plasma expansion because of non-linear character of equations and for the presence of an exponential term in the heating rate. With some approximation it is possible to find scaling laws only in the phase of strong laser absorption, that is for times t Ettrans. In this case the expression of H becomes

H B I0 mexp

k

g

h

l

where the Gaussian pulse reaches its maximum in t 4t *.

We can begin our calculations starting from eq. (16), which is now

3 2 d dt

g

h

k

g

h

l

and search for a solution of the kind

L 4L0tjexp [a(t 2t)2] ,

(34)

where L0, a, j and t are the parameters to be found.

We note that eq. (34) is valid only for initial times since it is physically clear that the plasma cannot regress.

Substituting the last equation in (33) with its derivatives, we find

L02

g

!

h

y

z

.

`

`

/

`

`

´

By equating the exponential terms in both members in eq. (35), we find t * 4t and

L.FERRARIOandD.BATANI

58

Fig. 7. – Numerical and analytic results (dashed and solid lines, respectively) are given by using a laser pulse showing a Gaussian intensity profile (about input parameters, see fig. 4).

all the terms included in the series. Thus, expression (35) reduces to the following form: 2 L2 0(j 21)(4j22 3 j) t2 j 234 2 I0 m (37)

and, as a consequence, we find j 43O2 and L04 ( 2 O 3 )kI0O m .

The final formula for the scalelength becomes

L(t) 4 2 3

o

y

g

h

z

while the evolution of temperature results from the momentum equation:

(39) _{T(t) 4} 2 3 cv I0 m t 3 O2_exp

y

g

h

z

{

}

where

.

`

/

`

´

The scaling law for temperature, compared with the numerical result, is represented in fig. 7. The comparison is significant only for early times. Hence effective methods are needed for the search of asymptotic solutions of hydrodynamic equations in all the phases of plasma expansion.

5. – 2D analytic solutions for special cases

We now go back to the more general case of plasma 2D expansion and we look for asymptotic laws for a cylindrical plasma expansion in some special cases: constant and laser heating and adiabatic expansion. In these cases, for the sake of simplicity, we suppose an irradiation of a thin film with a laser beam showing a flat-top (constant intensity) laser pulse.

5._{1. Constant heating. – We first consider the case in which t Et}transand hence H B

I0O m. We rewrite eqs. (4) in the following form:

Lx d 2 Lx dt2 4 2 3 cvT(t) , (41) Lr d 2_Lr dt2 4 2 3cvT(t) , (42) cv dT dt 4 2

k

g

h

g

h

l

Now, we multiply for ( 2 O3) and integrate the energy equation, and finally we substitute the momentum equations. Thus, we find

.

`

/

`

´

g

h

k

g

h

g

h

l

g

h

k

g

h

g

h

l

L.FERRARIOandD.BATANI

60

where a is a constant. By dividing by 2 the second equation of the set and summing to the first, we get

d dt

g

h

which, once integrated, becomes

Lx21 2 Lr24 2 3 Ht 3 1 2 cvT( 0 ) t21 Lx021 2 Lr02, (46)

where Lx0 and Lr0 represent the initial scalelength conditions, while T( 0 ) is related to

momentum equation d2_L

xO dt2N04 ( 2 O 3 ) cvT( 0 ) OLx0 ; besides, we assumed initial

velocities of the plasma’s expansion to be zero.

Since the initial value of Lx is small (Lx0bLr0), it fastly increases like t3 O2 (indeed

for H high enough, ( 2 O3) Ht3_{prevails on the second term). As a consequence, we can}

assume the following solution:

Lx4

o

2 9 Ht

3

1 Lx02 .

This is also implies LrB

k

Figure 8 shows the comparison between analytical results given by the formula concerning Lx and its corresponding numerical scalelength. In the numerical case we

find a transparency time ttransB 0 .58 ns.

After the plasma «forgets» the initial conditions, we find the asymptotic solutions for scalelength and temperature, governing the temporal evolution of the plasma for long times: L(t) 4 k2 3 H 1 O2_t3 O2_{, T(t) 4} 1 4 H cv t . (47)

It is interesting to notice that, though the kind of geometry used reduces the number of hydrodynamic equations because of symmetry around the axis of laser propagation, we recover the constants k_{2 O3 and 1O (4cv}) deduced by Hunter and London for a spherical expansion [5].

Figure 9 shows numerical results for scalelength on logarithmic scale; we can distinguish three regimes characterizing the plasma expansion (0D, 1D and 2D). The first one is the phase of plasma expansion in which the scalelengths are still dominated by the initial conditions, i.e. L(t) BL0 in all directions. In the second phase, the

expansion in the longitudinal direction becomes evident, while in the lateral direction it is still negligible, since L0 rcL0 x (being respectively of the order of the film thickness

and the laser focal spot). In this case the expansion is determined by the 1D solution already found by London and Rosen. This phase ends at a time of the order of t2 D.

Fig. 8. – Temporal course of analytical and numerical scalelengths. Main input parameters are: I 45.5Q1013

W Ocm2

, l 41.053 mm, aZb 48.4, aAb 418, W4400 mm, m43.3Q1024 _{g cm}22_{. The}

foil material is formvar.

5.2. Laser heating. – After the transparency time, the optical depth decreases and in a short time we can consider HBIatbOm. Thus, we can write the energy equation as

cv dT dt 4 2

k

g

h

g

h

l

where we used eq. (12) to write, in first approximation, atb.

It is not possible to find the exact analytic solution, but, as the numerical results show for large times, both scalelengths approach the same behaviour (L Pt). Thus the hydrodynamic set gives a solution for the temperature of the kind T(t) Pt28 O 5_{, typical}

of a 3D expansion.

5.3. Adiabatic expansion. – After the laser is turned off, the plasma is characterized by an adiabatic expansion with an electronic temperature going to zero for late times and an optical depth depending only on Lr(in fact, from eqs. (4), it is easy to show that T3 O2_L

xLr2 is a constant quantity).

Following the same considerations given for the case of constant heating, we get

Lx21 2 Lr24 3 at21 4 bt 1 4 c (48) with a 4LxFL O xF1 ( 1 O 3 )(L . 2 xF1 2 L . 2 rF), b 4 (1O2) LxFL . xF1 LrFL . rF2 ( 3 O 2 ) atF and c 4 ( 1 O4) L2

L.FERRARIOandD.BATANI

62

Fig. 9. – Course of scalelengths through the different regimes of expansion with constant heating. Both scalelengths approach the scaling laws given by eqs. (47). —— Lr, – – – Lx.

the quantity at the time tF, when the laser is turned off, while the dot denotes

differentiation with respect to time.

For large times, the system becomes spherical. Thus, the momentum equation takes the form

L d

2_L

dt2 B 0

(49)

pointing out a linear increase in time for the scalelength (L Pt), in accordance with eq. (48).

From the energy equation we find the following temporal dependence for the temperature:

T(t) 4 [L(t) ]22

P t22. (50)

The last expression agrees with the normal formula of the adiabatic expansion

TVg 21_{4constant, as can be deduced by considering g 4 5 O 3 and V P L}3_.

6. – Conclusion

We have generalized the 1D self-similar model developed by London and Rosen for laser-heated exploding foils. By using the similarity hypothesis, we have reduced the hydrodynamics system to a set of ordinary differential equations in cylindrical coordinates, finding the asymptotic scaling laws for special cases.

Besides, we found analytic and numerical solutions for a nearly-trapezoidal intensity profile of laser pulse, which improve remarkably the comparison with the results given by using a flat-top pulse during the early times, when the plasma’s expansion is still 1D. We also note that these solutions have a more general validity than just 1D case. Indeed even in bidimensional geometries, 2D effects become important only after the time t2 D. Hence, if t2 DD t0, our solution for n.t. pulses can be

used and improve the description of laser-plasma interaction in the early phase. We are now working on the comparison between appropriate experimental data and the predictions given by our self-similar model.

* * *

We gratefully acknowledge the discussion with D. GIULIETTI, Department of Physics of Pisa and M. MILANI, University of Milan. This work was performed under the auspices of the CNR contract No. 94.00705.CT02.115.27689.

R E F E R E N C E S

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