1. INTRODUCTION
Ultrashort ultraintense (USUI) laser pulses of duration less than 10 laser periods and intensity up to 1021 Wcm−2 have attracted much recent attention (Wilks & Kruer, Reference Wilks and Kruer1997; Sarkisov et al., Reference Sarkisov, Bychenkov, Tikhonchuk, Maksimchuk, Chen, Wagner, Mourou and Umstadter1997; Ditmire et al., Reference Ditmire, Zweiback, Vanovsky, Cowan, Hays and Wharton1999; Karsch et al., Reference Karsch, Habs, Schatz, Schramm, Thirolf, Meyer-ter-Vehn and Pukhov1999; Malka et al., Reference Malka, Fritzler, Lefebvre, Aleonard, Burgy, Chambaret, Chemin, Krushelnick, Malka, Mangles, Najmudin, Pittman, Rousseau, Scheurer, Walton and Dangor2002, Reference Malka and Fritzler2004; Roth et al., Reference Roth, Blazevic, Geissel, Schlegel, Cowan, Allen, Gauthier, Audebert, Fuchs, Meyer-ter-Vehn, Hegelich, Karsch and Pukhov2002, Reference Roth, Brambrink, Audebert, Blazevic, Clarke, Cobble, Geissel, Habs, Hegelich, Karsch, Ledingham, Neelz, Ruhl, Schlegel and Schreiber2005; Hegelich et al., Reference Hegelich, Karsch, Pretzler, Habs, Witte, Guenther, Allen, Blazewic, Fuchs, Gauthier, Geissel, Audebert, Cowan and Roth2002, Reference Hegelich, Albright, Audebert, Blazevic, Brambrink, Cobble, Cowan, Fuchs, Gauthier, Gautier, Geissel, Habs, Johnson, Karsch, Kemp, Letzring, Roth, Schramm, Schreiber, Witte and Fernández2005; Mangles et al., Reference Mangles, Murphy, Najmudin, Thomas, Collier, Dangor, Divall, Foster, Gallacher, Hooker, Jaroszynski, Langley, Mori, Norreys, Tsung, Viskup, Walton and Krushelnick2004, Reference Mangles, Walton, Najmudin, Dangor, Krushelnick, Malka, Manclossi, Lopes, Carias, Mendes and Dorchies2006; Geddes et al., Reference Geddes, Toth, Tilborg, Van Esarey, Schroeder, Bruhwiler, Nieter, Cary and Leemans2004; Faure et al., Reference Faure, Glinec, Pukhov, Kiselev, Gordienko, Lefebvre, Rousseau, Burgy and Malka2004; Kawata et al., Reference Kawata, Kong, Miyazaki, Miyauchi, Sonobe, Sakai, Nakajima, Masuda, Ho, Miyanaga, Limpouch and Andreev2005; Yu et al., Reference Yu, Xu, He, Yu, Ishiguro, Zhang and Wong2005; Xu et al., Reference Xu, Yu, Lu, Senecha, He, Shen, Qian, Li and Xu2005; Glowacz et al., Reference Glowacz, Hora, Badziak, Jablonski, Cang and Osman2006; Koyama et al., Reference Koyama, Adachi, Miura, Kato, Masuda, Watanabe, Ogata and Tanimoto2006; Lifshitz et al., Reference Lifschitz, Faure, Glinec, Malka and Mora2006). In such a pulse, a huge amount of electromagnetic energy is concentrated in such a tiny volume that its interaction with matter involves new space, time, and energy scales (Gibbon, Reference Gibbon2005; Mourou et al., Reference Mourou, Tajima and Bulanov2006). The interaction physics, such as collisionless and collisional scattering and absorption of laser light, of the short pulses can be quite different from that for the longer (~ 1 ps) pulses containing many light-wave periods (Yu & Shukla, Reference Yu and Shukla1978; Kruer, Reference Kruer1988; Lashinsky et al., Reference Lasinski, Langdon, Hatchett, Key and Tabak1999).
For USUI laser pulses of only a few wave periods, the pulse envelope, and the wave length are of similar order of magnitude in both space and time (Wilks & Kruer, Reference Wilks and Kruer1997; Gibbon, Reference Gibbon2005; Mourou et al., Reference Mourou, Tajima and Bulanov2006). The light pressure of such a steep-gradient pulse is extremely strong and is on extremely short time scale, and new phenomena can appear. For example, the ponderomotive or light pressure of an USUI laser pulse can drive away almost all the plasma electrons in the pulse region, leaving behind the massive ions and an intense space-charge force which balances the ponderomotive force (Gibbon, Reference Gibbon2005). A positively charged propagating electron cavity co-existing with the self-trapped light is thus created (Yu & Shukla, Reference Yu and Shukla1978). If the plasma is underdense, the cavity-pulse system will propagate through the slab without losing much energy. If the plasma is overdense, the laser will be mostly reflected at the vacuum-plasma interface. Between these two limits one can expect a regime where a part of the laser pulse would penetrate into the plasma, be trapped, and eventually stopped inside. In that case, highly localized intensive heating of a very small volume of the plasma can be realized.
In this paper, we investigate by particle-in-cell (PIC) simulation the interaction of an USUI laser pulse in a critical-density plasma by PIC simulation. It is shown that a significant part of the USUI pulse can indeed enter into and propagate a finite distance in the plasma, be trapped and completely stopped, and its energy efficiently deposited in a small plasma region as energetic ions.
In the next section, we present a brief description of the proposed process and a discussion of the differences between this and the related processes. In Section 3, we summarize the PIC simulation configuration and parameters. For the sake of comparison, we present and briefly discuss the results for several different interactions. In Section 4, we analyze the trapping and stopping of the transmitted light pulse for the case of interest, and in Section 5, we discuss the local plasma heating. In Section 6, the validity of our model and a discussion of the results and possible application to fast ignition are given.
2. INTERACTION OF USUI LASER WITH DENSE PLASMA
A laser pulse will propagate in a plasma if the electron density n satisfies n < γn c, where γ = √1 + aReference Deutsch2 (with a = eA/m ec, where e and m are the electron charge and rest mass, A is the laser-light vector potential, and c is the speed of light) is the relativistic factor. When an USUI pulse impinges on a critical-density (n ~ γn c) plasma, a part of the laser light is reflected, but a significant part can still enter the plasma as a smaller light pulse together with a number of much smaller light fragments originating from the impact. Despite the high density, the plasma electrons in the pulse region can be almost completely expelled by the ultraintense ponderomotive force (Yu & Shukla, Reference Yu and Shukla1978; Wilks & Kruer, Reference Wilks and Kruer1997; Kawata et al., Reference Kawata, Kong, Miyazaki, Miyauchi, Sonobe, Sakai, Nakajima, Masuda, Ho, Miyanaga, Limpouch and Andreev2005), resulting in an electron cavity with an intense space-charge field. At the cavity boundaries the ponderomotive and space-charge forces on the electrons balance. The propagation of the self-trapped light pulse will slow down when the local electron density at the front of the pulse-cavity system increases due to pile-up of the expelled plasma electrons, as well as decrease of the relativistic factor γ because of light dissipation. Eventually n approaches γn c and the tiny light and cavity system comes to a stop. Since the pulse is stationary, a longer time scale becomes relevant and processes such as Coulomb explosion (Sarkisov et al., Reference Sarkisov, Bychenkov, Tikhonchuk, Maksimchuk, Chen, Wagner, Mourou and Umstadter1997; Ditmire et al., Reference Ditmire, Zweiback, Vanovsky, Cowan, Hays and Wharton1999) of the cavity ions, formation of ion channel and shock layers, as well as their subsequent relaxation, occur. The trapped light energy is thus efficiently transferred to the plasma within a highly localized region in the plasma.
Despite several apparent similarities such as plasma channel formation, the process here is physically very different from that of a long-pulse (~ ps) laser in a dense plasma. There the ions can follow the ponderomotively driven electrons when the laser is still acting on the electrons, so that there is little or no generation of charge-separation field. Instead, hole digging (Kodama et al., Reference Kodama, Takahashi, Tanaka, Tsukamoto, Hashimoto, Kato and Mima1996, Reference Kodama, Tanaka, Yamanaka, Kato, Kitagawa, Fujita, Kanabe, Izumi, Takahashi, Habar, Okada, Iwata, Matsushita and Mima1999; Harbara et al., Reference Habara, Lancaster, Karsch, Murphy, Norreys, Evans, Borghesi, Romagnani, Zepf, King, Snavely, Akli, Zhang, Freeman, Hatchett, MacKinnon, Patel, Key, Stoeckl and Stephens2004) of the plasma by the laser occurs. Hole digging by laser is physically similar to self-consistent cavitation of discharge plasma by RF fields (Kim et al. Reference Kim, Stenzel and Wong1974; Yu & Shukla, Reference Yu and Shukla1978). The process studied here also differs from the interaction of an USUI pulse in a low-density plasma for producing monoenergetic particle bunches (Mangles et al., Reference Mangles, Murphy, Najmudin, Thomas, Collier, Dangor, Divall, Foster, Gallacher, Hooker, Jaroszynski, Langley, Mori, Norreys, Tsung, Viskup, Walton and Krushelnick2004; Geddes et al., Reference Geddes, Toth, Tilborg, Van Esarey, Schroeder, Bruhwiler, Nieter, Cary and Leemans2004; Faure et al., Reference Faure, Glinec, Pukhov, Kiselev, Gordienko, Lefebvre, Rousseau, Burgy and Malka2004). There a propagating electron cavity and/or wake plasma oscillation are created since d > λ and the condition (Wilks & Kruer, Reference Wilks and Kruer1997; Mourou et al., Reference Mourou, Tajima and Bulanov2006) d/λ <√n c/n 0, where d and λ are the laser pulse width and wavelength, n 0 and n c are the plasma and critical densities, for wake generation can be satisfied. The laser energy will then be mainly transferred to the wake-field, which can accelerate a small number of electrons almost mono-energetically. Finally, the present problem also differs from that of target-normal acceleration of surface ions in a solid-density plasma by an intense laser pulse (Karsch et al., Reference Karsch, Habs, Schatz, Schramm, Thirolf, Meyer-ter-Vehn and Pukhov1999; Hegelich et al., Reference Hegelich, Karsch, Pretzler, Habs, Witte, Guenther, Allen, Blazewic, Fuchs, Gauthier, Geissel, Audebert, Cowan and Roth2002; Roth et al., Reference Roth, Brambrink, Audebert, Blazevic, Clarke, Cobble, Geissel, Habs, Hegelich, Karsch, Ledingham, Neelz, Ruhl, Schlegel and Schreiber2005). In that case, the latter hardly penetrates the plasma and ion acceleration is accomplished by the space-charge field created by the laser-expelled electrons.
3. PARTICLE-IN-CELL SIMULATION
In this study, we are interested in the basic physical mechanism of laser pulse stopping. For simplicity we shall use a two-dimensional (2D) configuration for the PIC simulation. For the purpose of comparison, we shall present the results for the interaction of an USUI laser pulse with a plasma at below, near, and above the critical density (Wilks & Kruer, Reference Wilks and Kruer1997; Xu et al., Reference Xu, Yu, Lu, Senecha, He, Shen, Qian, Li and Xu2005). The simulation box (x, y) is 40λ0 × 10λ0, where λ0 = 1 µm is the laser wavelength. Inside the box, a 20λ0 long plasma slab is placed at the center, with 10λ0 long vacuum segments on both sides of the slab. A linearly polarized laser pulse with wave electric field E = E zêz enters the box from the left boundary (x = 0) and propagates through the left vacuum region into the plasma slab at x = 10λ. The pulse, Gaussian in the x and y directions, has a spot size w 0 = 3λ0 and FWHM d = 5λ0. The simulation grid contains 1200× 600 cells, with 25 particles per cell, and the minimum time step is 0.015T0, where T 0 is the light wave period (Xu et al., Reference Xu, Yu, Lu, Senecha, He, Shen, Qian, Li and Xu2005). The initial electron temperature is 1 keV, and the electron-ion mass ratio is 1/1836.
We first consider a laser pulse of strength parameter a = 10 and background density n 0 = 0.5n c (underdense plasma), where n c = 1021 cm−3. Figure 1 shows snapshots of |E z| at t = 21, 27, and 33 T 0. As expected, the pulse suffers very little (about 5%) reflection at the vacuum-plasma boundary and the plasma electrons are pushed aside by the radiation pressure. An electron cavity is created in the plasma and it moves together with the laser pulse. The self-trapped pulse-cavity system eventually passes through the plasma slab. Next, we consider the interaction of same laser pulse with a plasma at critical density (n 0 = n c). Figure 2 shows snapshots of |E z| at t = 24, 30, and 36 T 0. We see that reflection at the boundary is significant (~25%). As the transmitted pulse propagates in the plasma, it also create a co-moving electron cavity. The pulse also deforms significantly, especially in its tail part. The main pulse still contains much light energy. But before x = 20λ0 this self-trapped light pulse comes to a complete stop and remains there until its energy is totally dissipated. At still higher plasma densities, such as n 0 = 2n c in Figure 3, the same laser pulse is mostly (more than 60%) reflected at the slab boundary. In fact, the laser can only penetrate a short distance into the plasma and no closed electron cavity is formed. However, a sufficiently intense laser pulse can still penetrate into the n 0 = 2n c plasma and be stopped inside it, as shown in Figure 4 for a = 20. In this case, the interaction is similar to that for the a = 10, n 0 = n c interaction shown in Fig. 2), although here much higher energy is involved.
Fig. 1. (Color online) Snapshots of the longitudinal laser electric field |E z| (arb. units) at t = (a) 21 T 0, (b) 27 T 0, and (c) 33 T 0, for a = 10 and n 0 = 0.5n c. The laser pulse can pass through the plasma slab located in 10λ0 < x < 30λ0.
Fig. 2. (Color online) Snapshots of the longitudinal laser electric field |E z| at t = (a) 24T 0, (b) 30T 0, and (c) 36T 0, for a = 10 and n 0 = n c. The laser pulse is trapped inside the plasma slab.
Fig. 3. (Color online) Snapshots of the longitudinal laser electric field |E z| at t = (a) 15 T 0, (b) 18T 0, and (c) 21T 0, for a = 10 and n 0 = 2n c. Here the vacuum region on the left is also shown (recall that the vacuum-plasma boundary is at x = 10λ0). Thus, here most of the laser pulse is reflected and the remaining part can only penetrate a very short distance into the plasma.
Fig. 4. (Color online) Snapshots of the longitudinal laser electric field |E z| at t = (a) 20T 0, (b) 30T 0, and (c) 36T 0, for a = 20 and n 0 = 2n c. Similar to that in Fig. 2, the laser pulse can penetrate into the plasma slab and be stopped inside. Apparently the system is not more chaotic despite the higher interaction energy.
4. SELF-TRAPPING AND STOPPING OF LIGHT PULSE
To investigate in more detail the process of laser-light self-trapping, we now concentrate on the case a = 10 and n 0 = n c. Figure 5 shows the spatial distributions of (a) |E z|, (b) n, and (c) the scalar potential Φ at t = 24 T 0, which is approximately the time when the self-trapped pulse is stopped. The profiles of |E z|, n, and Φ at the pulse indicate that the light pressure pushes away the electrons, creating an electron-deficient cavity region. The laser light, weakened by energy loss to the electron cavitation process as well as density increase of the boundary plasma, becomes self-trapped and the pulse-cavity system propagates forward self-consistently until it is stopped near x = 20λ0, when the electron density in front of the pulse exceeds the effective critical density γn c. Figure 5b also shows that the electrons at the cavity boundary, and especially in the front of, the cavity are compressed to high density by the light pressure, and the electron density disturbance at the cavity boundary propagates into the unperturbed plasma. The highest density (~ 4n c, at x ~ 17 λ0) occurs at the pulse front where the compression of the electrons is strongest.
Fig. 5. (Color online) Snapshots of (a) |E z|, (b) electron density n, and (c) space-charge field Φ for the case n 0 = n c at t = 24T 0, which is just before the laser pulse comes to a complete stop.
5. ION DYNAMICS AND PLASMA HEATING
Electron cavitation with almost no ion participation can occur because the displacement of the massive ions accelerated by the space-charge and ponderomotive forces is insignificant on the initial short time scale. However, in the wake of the pulse or when the pulse is stopped, longer-time-scale effects such as Coulomb explosion (Sarkisov et al., Reference Sarkisov, Bychenkov, Tikhonchuk, Maksimchuk, Chen, Wagner, Mourou and Umstadter1997; Ditmire et al., Reference Ditmire, Zweiback, Vanovsky, Cowan, Hays and Wharton1999) of the ions in the cavity and ion channel formation in the wake plasma can occur.
Figure 6 shows the ion density n i at (a) 27T 0 (shortly after the pulse is stopped), (b) 42T 0, and (c) 57T 0. In (a) we see that except at the edge of the cavity where the relativistic ponderomotive force is strongest, the ion displacement is still insignificant. Figure 6b shows that Coulomb explosion of ions has started, ion holes and high density layers appear. In Figure 6c, we can see that an ion channel (where n i ≪ n 0) extending all the way back to the vacuum region is formed as the cavity ions are driven abruptly into the neighboring plasma, forming high-density shock-like layers at the channel edges. The channel has a rather long life-time since the inertial ions return only by collisional or turbulent diffusion. One can also notice that fine-structured wave-like ion-density patterns appear in the background plasma around and in front of the channel. The origin of these patterns is still unclear and deserves more detailed study. They could be a result of ion waves or wakes excited by the very energetic but small ion bunches that are also produced by the Coulomb explosion.
Fig. 6. (Color online) Snapshots of the ion density n i at (a) 25, (b) 42, and (c) 57T 0.
To confirm that the high-density layers corresponds to shocks (Sarkisov et al., Reference Sarkisov, Bychenkov, Tikhonchuk, Maksimchuk, Chen, Wagner, Mourou and Umstadter1997; Ditmire et al., Reference Ditmire, Zweiback, Vanovsky, Cowan, Hays and Wharton1999), in Figure 7 we present the ion energy (ion energy density divided by density) ɛi at (a) 27T 0, (b) 42T 0, and (c) 57T 0. One can see that the high-density and high-energy regions nearly overlap and they can thus be considered as shock layers. The ion energy in the latter can reach several tens KeV and density several times n 0.
Fig. 7. (Color online) Snapshots of the ion energy ɛi of the ions at (a) 25, (b) 42, and (c) 57T 0.
Figure 8 shows the evolution of the total (a) reflected laser energy (light energy in the left vacuum region minus the total injected laser energy, plus the light energy that has left the left computation boundary), (b) light energy in the plasma slab, and energy of the (c) electrons and (d) ions, all normalized by the incident laser energy. One can see that for n 0 = 0.5n c a laser pulse of a = 10 penetrates into the plasma with very little reflection. But the transmitted pulse can pass through the slab with only little energy. For n 0 = n c about 20% of the incident energy is reflected, and for n 0 = 2n c the reflection is more than 60%. The total light energy inside the plasma slab decreases with time, as the main electron cavity and other, smaller, structures are created. We can clearly see the two distinct time scales in the evolution of the laser, electron, and ion energies. One, the fast scale, is associated with the ponderomotively driven electron motion. The laser energy is rapidly spent in accelerating and compressing the electrons. The longer time scale is associated with the Coulomb expulsion of the cavity ions and the subsequent ion dynamics.
Fig. 8. (Color online) Time evolution of the total (a) reflected laser energy, (b) laser energy inside the plasma, and energy of the plasma (c) electrons and (d) ions, all normalized by the total incident laser energy, for n 0 = 0.5, 1, and 2.0n c, and a 0 = 10.
6. CONCLUSION
We have considered the 2D interaction of an USUI laser pulse with a dense plasma slab at near the critical density, such that the conditions for exciting wake plasma oscillations are not met. Despite the high density, a considerable fraction of an USUI laser pulse can enter the plasma slab, drive out the electrons, and form a cavity which traps the entered laser light. The light pulse-cavity system rapidly comes to a stop inside the plasma. In the resulting stationary cavity, Coulomb explosion of the ions creates a long-living plasma channel. The exploded ions have very high energy and form shock layers at the channel boundary. That is, in this process, the transmitted light energy is deposited within a very small stationary plasma region of shock-compressed density. Besides the main electron cavity and ion channel, the energetic laser impact also creates in the plasma different smaller structures and wave patterns that can be attributed to secondary cavities and electron or ion waves generated at the cavity and channel boundary. Some localized structures still contain light energy, but they do not seem to affect the evolution of the main cavity but can affect the heating and transport properties of the background plasma.
The possibility of laser-pulse trapping and stopping in a plasma indicates that one can focus a number of USUI laser pulses on a dense fuel plasma such that the penetrated light pulses are stopped at a common small plasma region and form either a single super cavity or many closely packed cavities. The location and intensity of the heating can be controlled by managing the laser intensity and the fuel density, as well as laser focusing. The multi-keV ion bunches resulting from the Coulomb explosion can efficiently heat a fuel plasma (Deutsch et al., Reference Deutsch, Furukawa, Mima, Murakami and Nishihara1997; Deutsch, Reference Deutsch2004; Liftshitz et al., Reference Lifschitz, Faure, Glinec, Malka and Mora2006), and may be superior to that from laser-cluster interaction which also depends on Coulomb exploded ions (Sarkisov et al., Reference Sarkisov, Bychenkov, Tikhonchuk, Maksimchuk, Chen, Wagner, Mourou and Umstadter1997; Ditmire et al., Reference Ditmire, Zweiback, Vanovsky, Cowan, Hays and Wharton1999) since here the energy deposition can be much more local and concentrated. Our 2D simulation precludes certain instabilities (Mourou et al., Reference Mourou, Tajima and Bulanov2006) and other higher dimensional effects, especially at longer times. Nevertheless, the results here should be useful as a starting point for laboratory experiments as well as more elaborate simulations. Finally, it may be of interest to point out that the interaction considered here is quite similar to that of USUI electron- or positron-bunch interaction with dense plasma (Deutsch et al., Reference Deutsch, Furukawa, Mima, Murakami and Nishihara1997; Deutsch, Reference Deutsch2004; Lu et al., Reference Lu, Huang, Zhou, Mori and Katsouleas2005; Zhou et al., Reference Zhou, He and Yu2006).
ACKNOWLEDGMENTS
This work was supported by the National High-Tech ICF Committee of China, the National Basic Research Program of China (973 Program), the National Science Foundation of China (Projects 10474081, 10676010, and 10476010), the International Foundation for Science, Health, and the Environment (USA), and the Max Planck Society - Chinese Academy of Sciences Exchange Program.
