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Absorption of ultra-short laser pulses and particle transport in dense targets

Published online by Cambridge University Press:  08 June 2006

M. SHERLOCK ,
A. R. BELL  and
W. ROZMUS
M. SHERLOCK
Affiliation:
Blackett Laboratory, Imperial College London, Prince Consort Rd, London, UK
A. R. BELL
Affiliation:
Blackett Laboratory, Imperial College London, Prince Consort Rd, London, UK
W. ROZMUS
Affiliation:
University of Alberta, Edmonton, Canada
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Abstract

A new version of the numerical code KALOS has been developed to solve the Vlasov-Fokker-Planck equation for electrons as well as EM wave propagation. KALOS represents the electron distribution function in momentum space by an expansion in spherical harmonics. Its unique features make possible simultaneous investigations of fast electron generation and transport and the collisional evolution of thermal particles, including the return current of cold electrons. We report here on results obtained in one spatial dimension. Absorption of 100fs, 1015 W/cm2 laser pulses has been studied at normal incidence in sharp-edged dense plasmas. We have studied the effect on absorption of energy transport into the target as well as the deviation of the electron distribution function from Maxwellian. It is shown that it is necessary to take into account collisional heat transport into the target in order to correctly model the absorption rate at the front surface.

Keywords

Laser-plasma interactionsTransport in high energy density plasmasVlasov-Fokker-Planck modeling of plasmas
Type
Research Article
Information
Laser and Particle Beams , Volume 24 , Issue 2 , June 2006 , pp. 231 - 234
Copyright
© 2006 Cambridge University Press

1. INTRODUCTION

The interaction of short laser pulses with solid density targets has been studied in numerous experiments and by theoretical and computational modeling (Honrubia et al., 2004; Isakov et al., 2005; Malka & Fritzler, 2004; Roth et al., 2005; Danson et al., 2005). The absorption of laser energy and transport of heat into dense plasmas is central to the understanding of such interactions. For moderate laser intensities, Iλ2 < 1017 Wμm2/cm2, hydrodynamic simulations (Price et al., 1995; Eidmann et al., 2000) have properly reproduced experimental measurements of the absorbed laser fraction. At higher laser fluxes, collisionless absorption mechanisms and the generation and transport of relativistic electrons become important, and therefore kinetic models have to be used to describe laser plasma coupling. We report in this paper the test case study of laser pulse absorption by plasma with a steep density profile using the new kinetic code, KALOS (Bell & Kingham, 2003). We have modeled absorption and electron transport at normal laser incidence, and maximum intensity I0 = 1015 W/cm2 in a high density Al plasma. Several kinetic effects related to the non-Maxwellian electron distribution function (f) and reduced thermal transport have been observed in the simulations already at this moderate intensity which was previously considered in the hydrodynamic description (Eidmann et al., 2000).

So far, absorption has been studied primarily with Particle-In-Cell codes (PIC) (Gibbon et al., 1999; Sakagami & Mima, 2004) which lack the ability to realistically model particle collisionality, due to the high computational cost of both the collision algorithms themselves and the need to resolve the tail population of the distribution function (that is, many times the thermal speed). Although the laser field may be of sufficiently high strength near the front of the target to render collisional absorption negligible, collisions always play an important role in determining absorption indirectly, since they determine the strength of the cold return current inside the target as well as the transport of heat into the target. The cold return current regulates the fast electron propagation (Bell & Kingham, 2003) while transport of heat away from the front surface modifies the absorption. Thus, absorption and transport are interrelated processes and any attempt to model the absorption must include the effects of collisional transport. Fokker-Planck (Bibi et al., 2004) codes based on a spherical harmonic expansion of the electron distribution function (f) have proved ideal for modeling transport when f is close to isotropic, since in this case only a few terms in the expansion are required. In order for a similar approach to be employed to model absorption, it is necessary to incorporate many terms in the spherical harmonic expansion, since f is not close to isotropic in the presence of strong fields. The numerical code KALOS solves the Vlasov-Fokker-Planck equation for f:

with f being represented by a spherical harmonic expansion of arbitrary order:

The full electron-ion Fokker-Planck collision operator is solved for cold ions, while electron-electron scattering is carried out only for the isotropic component of f, where they are the dominant collision mechanism. The code has already been successfully used to model anisotropic transport processes such as the resistive collimation of electron beams in solid targets (Bell & Kingham, 2003). We have coupled the basic KALOS algorithm with Maxwell's equations to model the EM pulse. No temporal averaging of f is performed, allowing processes which occur on the timescale of the laser period to be fully resolved.

2. SELF-CONSISTENT MODELING OF ABSORPTION AND TRANSPORT

The highest intensities for which collisional absorption remains an important contribution to target heating (that is, those around 1015 W/cm2) are particularly difficult for a Vlasov–Fokker–Planck code to model, since the conversion efficiency of laser energy into thermal plasma energy is relatively low, and this leads to high R = vosc /vth ratios (vosc is the peak oscillation speed of an electron in the laser field and vth is the electron thermal speed). A high R suggests a highly anisotropic velocity distribution for the electrons, essentially leading to a need to model a higher phase-space volume with an associated computational cost. Nevertheless, this regime (I0 = 1015 W/cm2) is taken as a starting point for the calculations presented in this paper, and the phase-space volume can be adequately modeled with 502 spherical harmonics in the expansion of f. We consider a solid Al target with an initial electron number density dependence on distance into the target (x) given by:

where n0 is chosen to be 10% of the critical density, n1 is the density of solid Al, x0 is set to one laser wavelength, and Δx is 5% of the laser wavelength. The atoms are assumed to be in their highest charge state (Z* = 13) and are initialized at 50 eV, corresponding to a value of R of around 4.

Figure 1 illustrates the main physical processes which are included in our model of a 1 μm wavelength, 100 fs pulse normally incident at 1015 W/cm2 onto a solid Al target. Laser light propagates from the left-hand boundary with constant intensity and is reflected at the critical density surface, indicated by the dashed vertical line. The standing wave profile is illustrated by the electric field normal to the target surface (Ey), which penetrates the dense plasma by a distance on the order of the collisionless skin depth (λsk = c/ωpe). Collisional absorption and plasma heating take place predominantly in the skin layer as seen in the temperature profiles at t = 23 fs and t = 54 fs. Heat conduction is responsible for the extension of the temperature profiles into the overdense region.

The electron density profile (black) and the transverse laser field (red). Electron temperature profiles are also shown at 23 fs (blue) and 54 fs (green) in units of the initial temperature T0 = 50eV.

Figure 2 shows the growth of the peak temperature in the absorption region. Until approximately t = 15 fs heating takes place in the skin layer without conduction losses to the target bulk (Rozmus et al., 1996). The initial increase of the temperature is well reproduced by homogeneous plasma heating theory which accounts for the change in f due to the so-called Langdon effect (Langdon, 1980; Matte et al., 1988). The large initial value of the parameter α = Z(vosc /vth)2 ∼ 200 leads to strong inverse bremsstrahlung heating which dominates electron-electron collisions and leads to the following isotropic distribution function in the heating region (Brunner & Valeo, 2002):

where Γ is the gamma function. This time-dependent fit is matched to homogeneous Fokker-Planck simulations but has also been obtained analytically (Bochkarev et al., 2004). With our model, it is possible to view f in the heating and transport regions, and this is shown in Figure 3. Figure 3a shows f in the heating region (just beyond the critical surface) at t = 19 fs, displaying the characteristic features of Eq. (1), that is, a reduced number of slow electrons and a reduced tail of energetic particles. For comparison, we also show a Maxwellian distribution function at the same temperature. If a local Maxwellian f is assumed during the heating process, the temperature increase will strongly deviate from the predictions of KALOS (cf. Fig. 2). At later times, the homogenous heating model overestimates the peak temperature because of the transport losses due to heat conduction into the dense plasma. For comparison, Figure 3b shows f is closer to Maxwellian in the transport region (around x = 0.47 μm) where the laser field is negligible and any distortions to f are a result of heat flow only. The heat flow in this region is found to be both non-Spitzer-Harm and nonlocal in nature.

The time evolution of the peak temperature. A comparison of KALOS results (blue), the solution to the homogenous heating model with the Langdon effect (red) and the heating of a Maxwellian plasma (green).

The electron distribution function (EDF) in (a) the absorption region where the temperature peaks and (b) beyond the skin layer where f is modified by the flow of heat rather than the laser field.

3. CONCLUSIONS

We have demonstrated the feasibility of modeling both absorption and transport with a Vlasov–Fokker–Planck code and that collisional kinetic effect are important for absorption and transport at moderate laser intensities (∼1015 Wcm−2). The current theoretical basis for describing such interactions needs to be extended to nonhomogenous plasmas and couple heat flow and absorption into one model. The computational model is currently being extended to study higher intensities.

References

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Figure 0

The electron density profile (black) and the transverse laser field (red). Electron temperature profiles are also shown at 23 fs (blue) and 54 fs (green) in units of the initial temperature T0 = 50eV.

Figure 1

The time evolution of the peak temperature. A comparison of KALOS results (blue), the solution to the homogenous heating model with the Langdon effect (red) and the heating of a Maxwellian plasma (green).

Figure 2

The electron distribution function (EDF) in (a) the absorption region where the temperature peaks and (b) beyond the skin layer where f is modified by the flow of heat rather than the laser field.