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Relativistic Vlasov simulation of intense fs laser pulse-matter interaction
25 September 1995 PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 205 (1995) 388-392
Relativistic Vlasov simulation of intense fs laser pulse-matter interaction H. Ruhl, P. Mulser Theoretische Quantenelektronik, TH-Darmstadt. Hochschulstrasse 4A, 64289 Darmstadt, Germany
Received 6 July 1995; accepted for publication 31 July 1995 Communicated by M. Porkolab
Abstract
The results of a relativistic Vlasov simulation of intense laser light-matter interaction are given. Detailed calculations of energy deposition in a short scale length target versus angle of incidence are presented. The degree of absorption is found to depend on the magnitude of a secular surface magnetic field. First order force terms in the electromagnetic fields are identified as the origin of particle jets whose energy exceeds the energy obtained by j x B-heating. The connection between the Brunel effect and anomalous skin effect is given. PACS: 52.40.Nk; 52.65.+z; 52.60.+h
In this Letter the collective interaction of ultrashort p-polarized laser pulses of irradiances Zh2 = 1016lOI* W cmm2 ,um2 with overdense plasmas are studied. At these intensities the target surface will be ionized in a few cycles of the light wave by field ionization and by collisions. As the electron quiver energy becomes relativistic, collisions are negligible and efficient light absorption is mainly due to collective effects. Recent experiments of Price et al. [ l] indicate indeed that the laser-plasma interaction adopts a universal behaviour for high irradiances and contrast ratios which can only be explained by a predominantly collective interaction of the laser pulse with matter. For a pulse length of a few laser cycles as considered here no substantial ion motion occurs and resonance absorption does not take place. Instead, the Brunel effect [ 21 is currently believed to represent an important absorption mechanism for p-polarized light. As an alternative method to frequently used PIC
codes [2-61 we decided to develop a Vlasov code [ 7-91. Vlasov simulations are time-consuming, however, as a compensation they prove to be almost noisefree. In addition, there are specific problems, e.g. kinetic wavebreaking [ 81 and the anomalous skin effect (this paper), which we found difficult to solve by PIC methods. To calculate collective absorption in condensed matter we assume that the target is fully ionized, the ions forming an immobile, overdense background separated from the vacuum by a small plane transition layer of relative thickness L/A < 1, A being the vacuum wavelength. The laser pulse is represented by a plane p-polarized wave. Thus, in case of oblique incidence a Lorentz boost of magnitude uY = c sin 8 can be performed which leads back to the situation of normal incidence [ lo]. In a numerical simulation such a transformation was used first by Gibbon and Bell [ 31. The electron distribution function fe (xp, pp ) is governed
0375-9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights resewed SSDIO375-9601(95)00596-X
H. Ruhl, P. Muiser/Physics
Letters A 205 (1995) 388-392
389
21.0
by the relativistic Vlasov equation (7 proper time),
F
8.00
:
dpc” a dr ap
dx” a
x-+--
along with the characteristics dxp dr
-=-
pw
dp”
-
m ’
dr
= - f$'p"p,
,
(lb)
with the r.h.s. term of the second equation representing the Minkowski force. The field tensors Ffi” = #‘A” - VA&, A” being the vector potential, and P@’ are obtained from Maxwell’s equations,
_-&FP= jY
(lc)
qc* ’
with the current source term j’ = j,” + j/’ where j," = -e
s
d4p cp’ x 20(pe)S(p2
-21.0,
-12.0
,
-6.0
i
,
0.0
6.0
12.0
lR.O
P&VU,
Fig. 1. Contour line plot of the cycle-averaged electron distribution function fe (x, pr. pY, t), centered at T = 8 cycles, at critical density in a logarithmic scale. The parameters are IA2 = 1018W cm-* prn2, n/& = 25, T, = 10 keV, L/A = 0.023, and O=O”.
- m*c*) fe.
The source term jr takes account of the background ion current in the boosted frame. The numerical solution of the system of equations (la) is accomplished by a newly developed splitting scheme. The size of the simulation box needed depends on the intensity and the scale length. Four vacuum wavelengths in the xdirection and 50rnc~, (Q, is the thermal velocity) in momentum space proved to be sufficient to conserve energy and particle number. First, collective absorption of a plane wave under normal incidence is studied. The ion density ni, normalized to the critical density n,, is taken as large as ni = 25. The electrons start from a Maxwellian distribution of initial temperature Te = 10 keV and a local equilibrium particle density n(x) corresponding to the ambipolar field produced by T,. In order to simulate the pure case of a solid target with a step-like transition from the vacuum n = 0 to n = 25 the transition layer was chosen as small as L/A = 0.023. The laser beam is switched on over half an oscillation period. Four cycles later all physical quantities have passed into a steady state. Fig. 1 shows the contour lines of the timeaveraged electron distribution function fe(x, px, py , t) at critical density x = xc. The perfect symmetry of fe with respect to pr = 0 suggests fast electron generation twice a cycle. Due to the conservation of the transverse canonical momentum the force on the electrons is second order in A, and oscillates with twice
10.5
<
0.0
a” -10.5
-21.0 -0.33
-019
0.00
0.19
n.33
0.57
x/h
Rg. 2. Contour line plot of the cycle-averagedelectron distribution
function ge( x,pY, t), centered at T = 8 cycles, in a logarithmic scale. The parameters are IA2 = IO’* W cm-2 pm2, n/n, = 25, Te = 10 keV, L/A = 0.023, and 8 = O”. The position of n, is indicated by xc.
the frequency of the laser field (see last paragraph). Hence, it produces jets twice per cycle as was first described by Kruer and Estabrook [ 111. Simple reasoning shows that their maximum energy E,, is given by
..,=mc*{ [l+pJJ1’*- I} -3lOkeV,
(2)
which equals four times the mean vacuum oscillation energy (Q, = ,&%r). A similar expression was obtained by Wilks [ 121. For normal incidence the transverse coordinate (y-coordinate) is cyclic. Hence the associated canonical momentum
H. Ruhl, P. Mulser/Physics Letters A 205 (1995) 388-392
390
16.0
9.0
$ a”
0.0
-9.0 -10.0i ,,,,,,,,,, -1.6
-0.8
XC I,,,,,,,,,,,,,,,,,,, 0.0
0.0
1.0
,,,] 2.4
3.2
Jh Fig. 3. Contour line plot of the electron distribution function ge( x, px, f) in a logarithmic scale at T = 9.54 cycles. The parameters am IA* = lo’* W cm-* pm*, n/nC = 2, Te = 10 keV, L/A = 0.15, and 0 = 0’. The position of k is indicated by xc. 15.00 F
8.00
Fig. 5. Contour line plot of the electron distribution function ge (x, px, t) in the boosted frame in a logarithmic scale at T = 9.54 cycles. The parameters am. IA* = lot* W cm-* pm*, n/k = 2, Te = 10 keV, L/A = 0.15, and 0 = 45O. The position of n, is indicated by xc.
I 62.5
4.25 50.0 :; c. L;
-17.25
-
-26.00
1 -12.0
2 s d e.
37.5
25.0
12.5 I -6.0
i 0.0
/ 6.0
I 12.0
1 18.0
P&V,, Fig. 4. Contour line plot of the cycle-averaged electron distribution function fe (n, px. pv, 1). centered at T = 8 cycles, in the boosted frame at critical density in a logarithmic scale. The parameters are IA* = 10’s W cm-* pm*, n/n, = 25, Te = 10 keV, L/A = 0 .023 , and 0 =45’.
pv - eA,( x, t) is conserved. Since A, vanishes in the overdense plasma the electron distribution function g,(x,p,, t) = s dp, fe(~,pxrpy,t) must retain its original shape there (see Fig. 2). For the parameters of Figs. 1 and 2 the formation of electron jets results in an absorption of 13.6%. The generation of fast particles is best seen from the time-resolved distribution function g,(x,p,,t) = s dp, fe(x,px,py,t). It is particularly pronounced for low n/n, values (see Fig. 3 for n/n, = 2). The case of oblique incidence is studied in the boosted frame in which the transverse canonical momentum py - e A, (x, t) is again conserved (boost velocity uy = cp where p = sin 0). Fig. 4 shows that
0.0 0.0
IS0
30.0
45.0
600
75.0
90.0
00 Fig. 6. Absorption versus angle of incidence. The parameters common to the bold curves are Te = 10 keV, n/nc = 25; the other parameters for theses curves are IA* = lOI W cm-* pm*, L/A = 0.023 (solid); IA* = lO** W cm-* pm*, L/A = 0.046 (chained-dashed); IA* = 10” W cm-* pm2, L/A = 0.023 (dashed). The parameters common to the rest of the lines are Te = 10 keV, n/n, = 2; the remaining parameters are IA* = lot8 W cm-*pm*, L/A = 0.15 (solid); IA* = 1016 W cm-* pm*, L/A = 0.15 (chained-dashed); IA* = 1016W cm-* pm*, L/A = 1.25 (dashed).
for an angle of incidence of 8 = 45” particle jets of two different peak velocities exist. The energy of the fastest particles now exceeds the energy of those generated at normal incidence. Analysis shows that the force on the electrons is now first order in A, (see last paragraph). Thus, it oscillates with the laser frequency and produces electron jets only once per cycle. This is the reason for the asymmetry of fe ( X,px, py , t) in Fig. 4. For the maximum energy, Eq. (2), we obtain
H. Ruhl, P. Mulser/Physics
emax 21 485 keV which is now six times the mean vacuum oscillation energy. The time-resolved distribution function ge(x,px, t) = s dp, fe(x,px,pr, t) for n/n, = 2 is shown in Fig. 5. Of particular interest is the cycle-averaged overall absorption coefficient A versus angle of incidence for different parameters (Fig. 6). It shows important aspects not observed in earlier simulations: (i) for short scale lengths maximum absorption occurs around 0 = 70’; (ii) for high irradiances substructures (secondary maximum and local minimum) are present; (iii) the degree of absorption correlates with the magnitude of a secular (dc) magnetic field generated on the target surface (compare Figs. 6 and 7). For IA* = lOI W cm-* pm* and L/A = 0.023 (bold dashed curves) B, is small (B, 6 8 MG). The corresponding absorption profile has a simple shape with only one maximum (A,, 2 56%). For Z/1*= 1Oi8W cm-*pm2 and L/A = 0.023 (bold solid curves) B, reaches up to 33 MG. The associated peak absorption decreases by almost a factor of two compared to the case where B, max= 8 MG. In addition, the absorption profile shows a pronounced plateau. Furthermore, by increasing the scale length to L/A = 0.046 the secular magnetic field more than doubles (B Z“ax - 85 MG), thus lowering overall absorption below the former values and depressing the former plateau to a pronounced minimum at 0 N 45”. To prove the assertion that the dc magnetic field controls the degree of abso tion we observe that the oscillation fields grow like je 1A2.Thus, their magnitude grows by a factor of three if the irradiation is increased tenfold. From Fig. 7 we see that the dc magnetic field grows by a factor of six from IA* = lOi W cm-* pm* to IA* = lOI W cm-* pm*. As a consequence the ratio between dc and oscillating fields doubles. The driving force on the electrons is given by F, M -e( E, cPBz ) for large angles of incidence. The electrostatic field E, can be neglected here since in the frame of normal incidence the laser pulse does not have a longitudinal electric field. Since the dc magnetic field is positive the force F, is weakened when it tries to pull electrons into the underdense plasma. Thus, fractional absorption decreases. Hence, it is mainly this dc magnetic field which causes the complexity of the absorption profile in the figure. Its origin is light pressure. The electron density is ponderomotively steepened and hence in the largely overdense plasma (n, = 25n,) a
Letters A 205 (1995) 388-392
391
Fig. 7. Peak values of secular surface magnetic field B, versus angle of incidence in the boosted frame. The parameters in common are n/k = 25 and Te = 10 keV. The other parameters are IA2 = lO’*W cm-* pm*, L/A = 0.023 (solid); IA* = lOI W cm-*pm*, L/A = 0.046 (chained-dashed); IA* = lOI W cm-* pm*, L/A = 0.023 (dashed). c:
‘0
1
5.00
1.2 :
P
8.00 1.0
3.75 .i W”
2.50
;* -?
1.25 -
-_)
-
0.8 0.6
G
0.4
_>w
0.00 -1.25)
, -0.38
-0.19
I.
:.
0:oo
--
,
,
0.19
0.36
- 0.” _ t , L 0.57
0.0
x/x
Fig. 8. Energy deposition j,& (solid) andjyEy (chained-dashed) versus position in arbitrary units in the lab frame. The dotted line gives the density n. The parameters are IA* = lOI* W cm-* pm2, n/nc = 25, Te = 10 keV, Lf A = 0.023, and B = 45O.
charge double layer is generated, positive (n, < ni ) at the vacuum boundary and negative (n, < ni) inside. In the system boosted at a speed csin 13the charge imbalance gives rise to a narrow current double layer which, according to Ampere’s law leads to a well localized dc magnetic field. In the slightly overdense plasma with (n, = 2n,) substructures appear in the charge double layer with the consequence of substructures in B. For Ih* = lOi W cm-* pm* and n/n, = 2 (chained-dashed and dashed curves) we observe the familiar scaling of peak absorption with a scale length according to (27rL/A)*i3 sin* B x 0.5 [ 131. For IA* = lOI W cm-* pm* and n/n, = 25 (bold dashed curve
392
H. Ruhl, P. hiulser / Physics Letters A 205 (1995) 388-392 1.6
positive but for large angles it changes sign periodically. For large angles of incidence it is thus responsible for the Brunel mechanism. We obtain F, M -e(E,
Fig. 9. Energy deposition jxEx (solid) and jy EY (chained-dashed) versus position in arbitrary units in the boosted frame. The dotted line gives the density n. The parameters am I@ = lOr8 W cmm2 pm2, n/n, = 25, Te = 10 keV, L/A = 0.023, and 0 =45O.
in Fig. 6) the angle of incidence for peak absorption found in our calculations agrees with linear theory while the degree of absorption predicted by linear calculations (see for instance Ref. [ 14] ) is smaller than the values of our simulations. Of significant difference is the case IA2 = lo’* W crnm2 pm2 and n/n, = 2 (thin solid curve in Fig. 6) where peak absorption is obtained for 8 = 20”. The explanation for this result is due to a dynamically generated effective electron scale length which considerably exceeds that of the ion background. Hence, the conditions for resonance absorption are fulfilled and the location of peak absorption agrees with that for resonance absorption. Finally, we want to rise the question how the Brunel effect and absorption by the anomalous skin effect are related to each other. Since the transverse canonical momentum is conserved we obtain for the force on the electrons Fx(7> =-e +
-cp+
[
E,(~(T),T) ~~(7) + &mc my3
>
1
&Ay(x(~) ~1 7
1
(3)
wherepy(r).=~y+e[AY(x(r),r)-Ay(x,t)l holds. For small angles of incidence this force is always
- cPB,)
= -iEk,
(4)
where E,” denotes the electrostatic field in the lab frame. As a consequence j,E,’ # 0 holds for 8 z+ 45”. Thus, j,E,’ gives the fraction of Brunel absorption while the fraction of skin absorption is given by RL (see Fig. 8). In the boosted frame j,E, disappears (see Fig. 9). Consequently, Brunel deposition and skin deposition cannot be distinguished from each other. Thus, the Brunel mechanism is a noncovariant energy deposition mechanism. Finally we want to point out that the linear version of the anomalous skin effect was recognized as a possible absorption mechanism as early as 1977 by Catto and More [ 151 and in 1988 it was first applied to fs laser pulse interaction
[161. References [I] D.F. Price, R.M. More, R.S. Walling, G. Guethlein, R.L. Shepherd, R.E. Steward and W.E.White, Phys. Rev. LA%, to be published (1995). [2] F. Brunei, Phys. Rev. Lea. 59 (1987) 52; Phys. Fluids 31 (1988) 2714. [3] I? Gibbon and A.R. Bell, Phys. Rev. Lett. 68 (1992) 1535. [4] P Gibbon, Phys. Rev. Lett. 73 (1994) 664. [5] SC. Wilks, W.L. Kruer, M. Tabak and A.B. Langdon, Phys. Rev. Lett. 69 (1992) 1383. 161 J. Denavit, Phys. Rev. Len. 69 (1992) 3052. [7] A. Bergmann, S. Huller, P Mulser and H. Schnabel, Europhys. L&t. 14 (1991) 661. [8] A. Bergmann and P Mulser, Phys. Rev. E 47 ( 1993) 3585. [9] B.N. Chichkov, Y. Kato, H. Ruhl and S.A. Uryupin, Phys. Rev. E 50 (1994) 2691. [lo] A. Boudier, Phys. Fluids 26 (1983) 1804. [ 111 W.L. Kruer and K. E&brook, Phys. Fluids 28 ( 1985) 431. [ 121 S.C. Wilks, Phys. Fluids B 5 (1993) 2603. [ 131 H.-J. Kull, Phys. Fluids 26 (1983) 1881. [ 141 A.A. Andreev et al., Sov. Phys. .JETP 74 (1992) 963. [IS] PJ. Catto and R.M. More, Phys. Fluids 20 (1977) 704. [ 161 P. Mulser, S. Pfalzner and F. Comolti, in: Proc. lnt. Workshop on Inertial confinement fusion, Varenna, 1988, ed. A. Caruso (Editrice Compositori, Bologna, 1989) p. 345.
ELSEVIER
LETTERS
A
Physics Letters A 205 (1995) 388-392
Relativistic Vlasov simulation of intense fs laser pulse-matter interaction H. Ruhl, P. Mulser Theoretische Quantenelektronik, TH-Darmstadt. Hochschulstrasse 4A, 64289 Darmstadt, Germany
Received 6 July 1995; accepted for publication 31 July 1995 Communicated by M. Porkolab
Abstract
The results of a relativistic Vlasov simulation of intense laser light-matter interaction are given. Detailed calculations of energy deposition in a short scale length target versus angle of incidence are presented. The degree of absorption is found to depend on the magnitude of a secular surface magnetic field. First order force terms in the electromagnetic fields are identified as the origin of particle jets whose energy exceeds the energy obtained by j x B-heating. The connection between the Brunel effect and anomalous skin effect is given. PACS: 52.40.Nk; 52.65.+z; 52.60.+h
In this Letter the collective interaction of ultrashort p-polarized laser pulses of irradiances Zh2 = 1016lOI* W cmm2 ,um2 with overdense plasmas are studied. At these intensities the target surface will be ionized in a few cycles of the light wave by field ionization and by collisions. As the electron quiver energy becomes relativistic, collisions are negligible and efficient light absorption is mainly due to collective effects. Recent experiments of Price et al. [ l] indicate indeed that the laser-plasma interaction adopts a universal behaviour for high irradiances and contrast ratios which can only be explained by a predominantly collective interaction of the laser pulse with matter. For a pulse length of a few laser cycles as considered here no substantial ion motion occurs and resonance absorption does not take place. Instead, the Brunel effect [ 21 is currently believed to represent an important absorption mechanism for p-polarized light. As an alternative method to frequently used PIC
codes [2-61 we decided to develop a Vlasov code [ 7-91. Vlasov simulations are time-consuming, however, as a compensation they prove to be almost noisefree. In addition, there are specific problems, e.g. kinetic wavebreaking [ 81 and the anomalous skin effect (this paper), which we found difficult to solve by PIC methods. To calculate collective absorption in condensed matter we assume that the target is fully ionized, the ions forming an immobile, overdense background separated from the vacuum by a small plane transition layer of relative thickness L/A < 1, A being the vacuum wavelength. The laser pulse is represented by a plane p-polarized wave. Thus, in case of oblique incidence a Lorentz boost of magnitude uY = c sin 8 can be performed which leads back to the situation of normal incidence [ lo]. In a numerical simulation such a transformation was used first by Gibbon and Bell [ 31. The electron distribution function fe (xp, pp ) is governed
0375-9601/95/$09.50 @ 1995 Elsevier Science B.V. All rights resewed SSDIO375-9601(95)00596-X
H. Ruhl, P. Muiser/Physics
Letters A 205 (1995) 388-392
389
21.0
by the relativistic Vlasov equation (7 proper time),
F
8.00
:
dpc” a dr ap
dx” a
x-+--
along with the characteristics dxp dr
-=-
pw
dp”
-
m ’
dr
= - f$'p"p,
,
(lb)
with the r.h.s. term of the second equation representing the Minkowski force. The field tensors Ffi” = #‘A” - VA&, A” being the vector potential, and P@’ are obtained from Maxwell’s equations,
_-&FP= jY
(lc)
qc* ’
with the current source term j’ = j,” + j/’ where j," = -e
s
d4p cp’ x 20(pe)S(p2
-21.0,
-12.0
,
-6.0
i
,
0.0
6.0
12.0
lR.O
P&VU,
Fig. 1. Contour line plot of the cycle-averaged electron distribution function fe (x, pr. pY, t), centered at T = 8 cycles, at critical density in a logarithmic scale. The parameters are IA2 = 1018W cm-* prn2, n/& = 25, T, = 10 keV, L/A = 0.023, and O=O”.
- m*c*) fe.
The source term jr takes account of the background ion current in the boosted frame. The numerical solution of the system of equations (la) is accomplished by a newly developed splitting scheme. The size of the simulation box needed depends on the intensity and the scale length. Four vacuum wavelengths in the xdirection and 50rnc~, (Q, is the thermal velocity) in momentum space proved to be sufficient to conserve energy and particle number. First, collective absorption of a plane wave under normal incidence is studied. The ion density ni, normalized to the critical density n,, is taken as large as ni = 25. The electrons start from a Maxwellian distribution of initial temperature Te = 10 keV and a local equilibrium particle density n(x) corresponding to the ambipolar field produced by T,. In order to simulate the pure case of a solid target with a step-like transition from the vacuum n = 0 to n = 25 the transition layer was chosen as small as L/A = 0.023. The laser beam is switched on over half an oscillation period. Four cycles later all physical quantities have passed into a steady state. Fig. 1 shows the contour lines of the timeaveraged electron distribution function fe(x, px, py , t) at critical density x = xc. The perfect symmetry of fe with respect to pr = 0 suggests fast electron generation twice a cycle. Due to the conservation of the transverse canonical momentum the force on the electrons is second order in A, and oscillates with twice
10.5
<
0.0
a” -10.5
-21.0 -0.33
-019
0.00
0.19
n.33
0.57
x/h
Rg. 2. Contour line plot of the cycle-averagedelectron distribution
function ge( x,pY, t), centered at T = 8 cycles, in a logarithmic scale. The parameters are IA2 = IO’* W cm-2 pm2, n/n, = 25, Te = 10 keV, L/A = 0.023, and 8 = O”. The position of n, is indicated by xc.
the frequency of the laser field (see last paragraph). Hence, it produces jets twice per cycle as was first described by Kruer and Estabrook [ 111. Simple reasoning shows that their maximum energy E,, is given by
..,=mc*{ [l+pJJ1’*- I} -3lOkeV,
(2)
which equals four times the mean vacuum oscillation energy (Q, = ,&%r). A similar expression was obtained by Wilks [ 121. For normal incidence the transverse coordinate (y-coordinate) is cyclic. Hence the associated canonical momentum
H. Ruhl, P. Mulser/Physics Letters A 205 (1995) 388-392
390
16.0
9.0
$ a”
0.0
-9.0 -10.0i ,,,,,,,,,, -1.6
-0.8
XC I,,,,,,,,,,,,,,,,,,, 0.0
0.0
1.0
,,,] 2.4
3.2
Jh Fig. 3. Contour line plot of the electron distribution function ge( x, px, f) in a logarithmic scale at T = 9.54 cycles. The parameters am IA* = lo’* W cm-* pm*, n/nC = 2, Te = 10 keV, L/A = 0.15, and 0 = 0’. The position of k is indicated by xc. 15.00 F
8.00
Fig. 5. Contour line plot of the electron distribution function ge (x, px, t) in the boosted frame in a logarithmic scale at T = 9.54 cycles. The parameters am. IA* = lot* W cm-* pm*, n/k = 2, Te = 10 keV, L/A = 0.15, and 0 = 45O. The position of n, is indicated by xc.
I 62.5
4.25 50.0 :; c. L;
-17.25
-
-26.00
1 -12.0
2 s d e.
37.5
25.0
12.5 I -6.0
i 0.0
/ 6.0
I 12.0
1 18.0
P&V,, Fig. 4. Contour line plot of the cycle-averaged electron distribution function fe (n, px. pv, 1). centered at T = 8 cycles, in the boosted frame at critical density in a logarithmic scale. The parameters are IA* = 10’s W cm-* pm*, n/n, = 25, Te = 10 keV, L/A = 0 .023 , and 0 =45’.
pv - eA,( x, t) is conserved. Since A, vanishes in the overdense plasma the electron distribution function g,(x,p,, t) = s dp, fe(~,pxrpy,t) must retain its original shape there (see Fig. 2). For the parameters of Figs. 1 and 2 the formation of electron jets results in an absorption of 13.6%. The generation of fast particles is best seen from the time-resolved distribution function g,(x,p,,t) = s dp, fe(x,px,py,t). It is particularly pronounced for low n/n, values (see Fig. 3 for n/n, = 2). The case of oblique incidence is studied in the boosted frame in which the transverse canonical momentum py - e A, (x, t) is again conserved (boost velocity uy = cp where p = sin 0). Fig. 4 shows that
0.0 0.0
IS0
30.0
45.0
600
75.0
90.0
00 Fig. 6. Absorption versus angle of incidence. The parameters common to the bold curves are Te = 10 keV, n/nc = 25; the other parameters for theses curves are IA* = lOI W cm-* pm*, L/A = 0.023 (solid); IA* = lO** W cm-* pm*, L/A = 0.046 (chained-dashed); IA* = 10” W cm-* pm2, L/A = 0.023 (dashed). The parameters common to the rest of the lines are Te = 10 keV, n/n, = 2; the remaining parameters are IA* = lot8 W cm-*pm*, L/A = 0.15 (solid); IA* = 1016 W cm-* pm*, L/A = 0.15 (chained-dashed); IA* = 1016W cm-* pm*, L/A = 1.25 (dashed).
for an angle of incidence of 8 = 45” particle jets of two different peak velocities exist. The energy of the fastest particles now exceeds the energy of those generated at normal incidence. Analysis shows that the force on the electrons is now first order in A, (see last paragraph). Thus, it oscillates with the laser frequency and produces electron jets only once per cycle. This is the reason for the asymmetry of fe ( X,px, py , t) in Fig. 4. For the maximum energy, Eq. (2), we obtain
H. Ruhl, P. Mulser/Physics
emax 21 485 keV which is now six times the mean vacuum oscillation energy. The time-resolved distribution function ge(x,px, t) = s dp, fe(x,px,pr, t) for n/n, = 2 is shown in Fig. 5. Of particular interest is the cycle-averaged overall absorption coefficient A versus angle of incidence for different parameters (Fig. 6). It shows important aspects not observed in earlier simulations: (i) for short scale lengths maximum absorption occurs around 0 = 70’; (ii) for high irradiances substructures (secondary maximum and local minimum) are present; (iii) the degree of absorption correlates with the magnitude of a secular (dc) magnetic field generated on the target surface (compare Figs. 6 and 7). For IA* = lOI W cm-* pm* and L/A = 0.023 (bold dashed curves) B, is small (B, 6 8 MG). The corresponding absorption profile has a simple shape with only one maximum (A,, 2 56%). For Z/1*= 1Oi8W cm-*pm2 and L/A = 0.023 (bold solid curves) B, reaches up to 33 MG. The associated peak absorption decreases by almost a factor of two compared to the case where B, max= 8 MG. In addition, the absorption profile shows a pronounced plateau. Furthermore, by increasing the scale length to L/A = 0.046 the secular magnetic field more than doubles (B Z“ax - 85 MG), thus lowering overall absorption below the former values and depressing the former plateau to a pronounced minimum at 0 N 45”. To prove the assertion that the dc magnetic field controls the degree of abso tion we observe that the oscillation fields grow like je 1A2.Thus, their magnitude grows by a factor of three if the irradiation is increased tenfold. From Fig. 7 we see that the dc magnetic field grows by a factor of six from IA* = lOi W cm-* pm* to IA* = lOI W cm-* pm*. As a consequence the ratio between dc and oscillating fields doubles. The driving force on the electrons is given by F, M -e( E, cPBz ) for large angles of incidence. The electrostatic field E, can be neglected here since in the frame of normal incidence the laser pulse does not have a longitudinal electric field. Since the dc magnetic field is positive the force F, is weakened when it tries to pull electrons into the underdense plasma. Thus, fractional absorption decreases. Hence, it is mainly this dc magnetic field which causes the complexity of the absorption profile in the figure. Its origin is light pressure. The electron density is ponderomotively steepened and hence in the largely overdense plasma (n, = 25n,) a
Letters A 205 (1995) 388-392
391
Fig. 7. Peak values of secular surface magnetic field B, versus angle of incidence in the boosted frame. The parameters in common are n/k = 25 and Te = 10 keV. The other parameters are IA2 = lO’*W cm-* pm*, L/A = 0.023 (solid); IA* = lOI W cm-*pm*, L/A = 0.046 (chained-dashed); IA* = lOI W cm-* pm*, L/A = 0.023 (dashed). c:
‘0
1
5.00
1.2 :
P
8.00 1.0
3.75 .i W”
2.50
;* -?
1.25 -
-_)
-
0.8 0.6
G
0.4
_>w
0.00 -1.25)
, -0.38
-0.19
I.
:.
0:oo
--
,
,
0.19
0.36
- 0.” _ t , L 0.57
0.0
x/x
Fig. 8. Energy deposition j,& (solid) andjyEy (chained-dashed) versus position in arbitrary units in the lab frame. The dotted line gives the density n. The parameters are IA* = lOI* W cm-* pm2, n/nc = 25, Te = 10 keV, Lf A = 0.023, and B = 45O.
charge double layer is generated, positive (n, < ni ) at the vacuum boundary and negative (n, < ni) inside. In the system boosted at a speed csin 13the charge imbalance gives rise to a narrow current double layer which, according to Ampere’s law leads to a well localized dc magnetic field. In the slightly overdense plasma with (n, = 2n,) substructures appear in the charge double layer with the consequence of substructures in B. For Ih* = lOi W cm-* pm* and n/n, = 2 (chained-dashed and dashed curves) we observe the familiar scaling of peak absorption with a scale length according to (27rL/A)*i3 sin* B x 0.5 [ 131. For IA* = lOI W cm-* pm* and n/n, = 25 (bold dashed curve
392
H. Ruhl, P. hiulser / Physics Letters A 205 (1995) 388-392 1.6
positive but for large angles it changes sign periodically. For large angles of incidence it is thus responsible for the Brunel mechanism. We obtain F, M -e(E,
Fig. 9. Energy deposition jxEx (solid) and jy EY (chained-dashed) versus position in arbitrary units in the boosted frame. The dotted line gives the density n. The parameters am I@ = lOr8 W cmm2 pm2, n/n, = 25, Te = 10 keV, L/A = 0.023, and 0 =45O.
in Fig. 6) the angle of incidence for peak absorption found in our calculations agrees with linear theory while the degree of absorption predicted by linear calculations (see for instance Ref. [ 14] ) is smaller than the values of our simulations. Of significant difference is the case IA2 = lo’* W crnm2 pm2 and n/n, = 2 (thin solid curve in Fig. 6) where peak absorption is obtained for 8 = 20”. The explanation for this result is due to a dynamically generated effective electron scale length which considerably exceeds that of the ion background. Hence, the conditions for resonance absorption are fulfilled and the location of peak absorption agrees with that for resonance absorption. Finally, we want to rise the question how the Brunel effect and absorption by the anomalous skin effect are related to each other. Since the transverse canonical momentum is conserved we obtain for the force on the electrons Fx(7> =-e +
-cp+
[
E,(~(T),T) ~~(7) + &mc my3
>
1
&Ay(x(~) ~1 7
1
(3)
wherepy(r).=~y+e[AY(x(r),r)-Ay(x,t)l holds. For small angles of incidence this force is always
- cPB,)
= -iEk,
(4)
where E,” denotes the electrostatic field in the lab frame. As a consequence j,E,’ # 0 holds for 8 z+ 45”. Thus, j,E,’ gives the fraction of Brunel absorption while the fraction of skin absorption is given by RL (see Fig. 8). In the boosted frame j,E, disappears (see Fig. 9). Consequently, Brunel deposition and skin deposition cannot be distinguished from each other. Thus, the Brunel mechanism is a noncovariant energy deposition mechanism. Finally we want to point out that the linear version of the anomalous skin effect was recognized as a possible absorption mechanism as early as 1977 by Catto and More [ 151 and in 1988 it was first applied to fs laser pulse interaction
[161. References [I] D.F. Price, R.M. More, R.S. Walling, G. Guethlein, R.L. Shepherd, R.E. Steward and W.E.White, Phys. Rev. LA%, to be published (1995). [2] F. Brunei, Phys. Rev. Lea. 59 (1987) 52; Phys. Fluids 31 (1988) 2714. [3] I? Gibbon and A.R. Bell, Phys. Rev. Lett. 68 (1992) 1535. [4] P Gibbon, Phys. Rev. Lett. 73 (1994) 664. [5] SC. Wilks, W.L. Kruer, M. Tabak and A.B. Langdon, Phys. Rev. Lett. 69 (1992) 1383. 161 J. Denavit, Phys. Rev. Len. 69 (1992) 3052. [7] A. Bergmann, S. Huller, P Mulser and H. Schnabel, Europhys. L&t. 14 (1991) 661. [8] A. Bergmann and P Mulser, Phys. Rev. E 47 ( 1993) 3585. [9] B.N. Chichkov, Y. Kato, H. Ruhl and S.A. Uryupin, Phys. Rev. E 50 (1994) 2691. [lo] A. Boudier, Phys. Fluids 26 (1983) 1804. [ 111 W.L. Kruer and K. E&brook, Phys. Fluids 28 ( 1985) 431. [ 121 S.C. Wilks, Phys. Fluids B 5 (1993) 2603. [ 131 H.-J. Kull, Phys. Fluids 26 (1983) 1881. [ 141 A.A. Andreev et al., Sov. Phys. .JETP 74 (1992) 963. [IS] PJ. Catto and R.M. More, Phys. Fluids 20 (1977) 704. [ 161 P. Mulser, S. Pfalzner and F. Comolti, in: Proc. lnt. Workshop on Inertial confinement fusion, Varenna, 1988, ed. A. Caruso (Editrice Compositori, Bologna, 1989) p. 345.