3. SIMULATION RESULTS

In order to study particle acceleration with an intense laser pulse, by varying the amplitude of laser pulse a = 1.0,1.5,2.0 and 3.0 (corresponding intensities I ≈ 1.37 × 10¹⁸, 3.09 × 10¹⁸, 5.49 × 10¹⁸, and 1.24 × 10¹⁹ W/cm² for a 1 μm laser pulse, respectively) only, and keeping other simulation parameters fixed, different PIC runs are performed. As in Figure 1, the electron energy distributions, from our simulations show, the maximum obtained electron net momentum goes up to p^max/m_e c ≈ 91, 151, 219, and 246, corresponding to maximum electron energy: E^max ≈ 46, 77, 112, and 126 MeV, respectively. An electron interacting with a plane electromagnetic wave (laser field) acquires the energy equal to E_e = m_e c²a²/2, for example, a = 2.0, E_e ≈ 1.0 MeV. From this point, one cannot expect that energetic electrons found in laser-plasmas are directly accelerated by an ultra-intense laser field.

Snapshots of electron energy distributions from 1D-PIC simulations. The plasma density is n = 0.01ncr.

The self-modulated regime of laser plasma interaction is defined by a laser pulse with a normalized vector potential of the order of unity a = eE₀ /m_eω₀ c ≈ 1, pulse length of several plasma periods, and the laser frequency several times the plasma frequency. In the one-dimensional limit, the interaction between such intense laser pulses and a low-density plasma is dominated by the forward Raman scattering (FRS) instability (Esarey et al., 1994; Mori et al., 1994; Mori, 1997). FRS modulates intense laser pulse at the plasma frequency with a corresponding modulation of the plasma density. The resulting large electron plasma wave grows until it reaches the wave-breaking limit, after which, large number of background electrons are self-trapped and accelerated to high energies. The self-modulated laser wakefield accelerator (SM-LWFA) utilizes this mechanism to generate a beam of highly energetic electrons.

The large amplitude plasma waves are generated by a self-modulation or forward Raman scattering instability with a phase velocity v_p, near the group velocity v_g of the laser pulse, v_p ≈ v_g, so γ_p = (1 − v_p²/c²)^−1/2 ≈ (1 − v_g²/c²)^−1/2 ≈ ω₀ /ω_p, where γ_p is the Lorentz factor corresponding to the phase velocity v_p of the plasma wave. According to Tajima and Dawson (1979), in the two-dimensional cold plasma theory, the critical amplitude of the plasmon is determined by the wave-breaking limit (standard dephasing limit), the maximum energy of an electron, trapped by the potential of a relativistic plasma wave, having an amplitude eφ^max ≈ γ_p m_e c² is given by E^max = 2γ_p²m_e c². By taking the parameter ω₀ /ω_p = 10 we used, the maximum energy of an electron obtained is E^max = 102 MeV. In two cases of a laser pulse, a = 1.0 and 1.5, the maximum relativistic factor and the electron energy obtained, are γ^max ≈ 91,151 and E^max ≈ 46,77 MeV, respectively, which is less than 102 MeV. This means that the large amplitude plasma waves do not exceed the wave breaking limit (standard dephasing limit). However, as the amplitude of laser pulse increase to a = 2.0 and 3.0, the maximum relativistic factor and the electron energy obtained are γ^max ≈ 219,246 and E^max ≈ 112,126 MeV, respectively, the maximum electron energy is exceeding the standard dephasing limit. It was reported in experiments (Ting et al., 1996; Wagner et al., 1997) and studied by PIC simulations (Tzeng et al., 1997; Gordon et al., 1998) that maximum relativistic electron energies observed are in an excess of the standard dephasing limit E^max = 2γ_p²m_e c². In recent years, a large effort has been made to understand why the maximum electron energy can exceed the standard dephasing limit. Studing the thermal effects on relativistic wave breaking has become an important issue (Sheng & Mayer-ter-Vehn, 1997). For example, Esarey et al. (1998), by including the space-charge field due to self-channeling, it was shown that the maximum energy E^max = 4γ_p²m_e c² gain is greater by a factor of ≥2. A new kind of mechanism that leads to efficient acceleration of electrons in underdense plasma by two counter-propagating laser pulses may help to explain how the maximum energy can possibly exceed the standard dephasing limit, for particle acceleration from the wave breaking, as suggested by some authors (Shvets et al., 1998; Sheng et al., 2002).

In addition to the above simulation runs, by changing the plasma length to L = 500, 1000, 1500c/ω₀, and fixing the laser amplitude to a = 1.0 and other plasma parameters, we get the maximum relativistic factor γ^max ≈ 41,91,134, with the corresponding maximum electron energy E^max ≈ 21,47,68 MeV, respectively. A fact is found, that is, the longer the plasma length, the larger the Lorentz factor γ_max.

In the next part, we leave the dephasing limit problem; our goal is to focus on a generation of a pulsed high-quality, well-collimated return electron beam, and to study ultra-relativistic beam characteristics, the mechanism of the formation, and the time-history in x and p_x space of the beam electrons.

In Figure 2, for the electron longitudinal momentum snapshots in phase-space x − p_x, physics of electron acceleration can be seen more clearly and directly. At an early stage background, electrons start to interact with a forward propagating relativistic electron plasma wave generated by a laser pulse, and are accelerated to large positive momenta p_x. The accelerated electrons move forward to the right plasma-vacuum boundary, as shown in Figures 2 and 3, and escape plasma layer into vacuum region to build a potential barrier, a large electrostatic field that is quickly formed (Fig. 4). This sheath ES field structure gradually decreases in vacuum region. The accelerated electrons entering this region experience deceleration to be eventually stopped. Further, by reflection, electrons reverse their motion to be accelerated backward into the bulk plasma. However, some ultra-fast electrons overcome the potential barrier and eventually are lost in the numerical-damping region, that we introduced.

Total electron momentum snapshots, in x − px phase-space from PIC simulation, for laser pulse, a = 1.0.

Test electron momentum snapshots, in x − px phase-space from PIC simulation, for laser pulse, a = 1.0.

Snapshots of electrostatic field Ex (normalized to mω0 c/e), in the case of laser pulse, a = 1.0.

Electrons travelling backward through the bulk plasma will be additionally accelerated by the ES sheath at the front interface (Fig. 4). As the time progresses, a mono-energetic well-collimated relativistic electron beam bunch (energy about 12–17 MeV) is eventually formed. When these electrons get through the front interface into vacuum, they again experience deceleration and reverse acceleration by the ES sheath potential. Most of the electrons returning into the bulk plasma are moving forward to the rear boundary. However, in the meantime, the high-quality well-collimated relativistic electron beamlet will gradually separate from the rest of the beam. Since the ES field at the plasma-vacuum boundary still exists, electrons which arrive at the rear side experience the similar process as before. Moving back and forth energetic electrons recirculate through the bulk plasma. In time, due to beam interactions with the bulk, relativistic beam features will eventually be lost via thermalization with the rest of the heated bulk plasma.

To see this process more clearly, by taking the time interval between 239 ≤ t ≤ 955 (corresponds to start and the end of this high-quality, well-collimated return electron beam), we select test electrons having negative momenta between −20 ≤ p_x /m_e c ≤ −90 in this time interval 239 ≤ t ≤ 955. About 1.3% of the total number of electrons satisfy this condition in our simulation. This means that, at least 1.3% electrons are accelerated to γ ≥ 20 in the case of a laser pulse with a = 1.0. These test electrons cover the whole range of the accelerated electrons that we discussed above. By introducing test electrons, we can trace back their earlier time history easily and vice versa. Figure 3 gives the test electron longitudinal momentum snapshots in phase-space x − p_x; a picture for electron acceleration, deceleration, forward and backward motion and so on, is more clear.

For the plasma length of 1000c/ω₀, laser pulse propagates in a plasma and arrives at the right plasma-vacuum interface at t ≈ 159; after that, although the laser pulse still propagates in a plasma, the electron acceleration by large plasma wave is almost finished by this time. Nearly all of the forward accelerated electrons are accelerated by the transient relativistic plasma wave before a laser pulse arrives at the right plasma boundary.

Figure 5 shows the snapshots of electron longitudinal momentum p_x distribution, for laser pulse a = 1.0. The peaked distributions can tell us about the formation of the highly-relativistic electron beam and its energy distribution in the opposite direction to the laser propagation. The beam is created as the accelerated electrons are reflected and pulled back into the bulk plasma at the rear plasma-vacuum interface. In the third snapshot (Fig. 5), apart from the highly-relativistic beam, also the thermalized, momentum peak (p_x ≈ 20) is observed; because some accelerated electrons have already returned (recirculate) to the bulk from the rear boundary after one cycle. As in Figure 6, the angular energy distribution θ ∼ tan⁻¹(p_y /p_x) of test electrons shows, the initial number of electrons which will be accelerated, which has angular energy distribution −90°–90° (moving forward, initially) is greater than that of 90°–270° (moving backward, initially). When accelerated in time, the electron angular distribution is mostly concentrated around two angles 90° and −270°, that is, |p_x| >> |p_y|; it demonstrates that electrons are accelerated mainly by the longitudinal waves.

Snapshots of longitudinal momentum distribution for incident laser pulse, a = 1.0.

Angular Energy-distribution θ ∼ tan−1(py /px) snapshots in the case of laser pulse, a = 1.0.

In order to better visualize, electron orbits in space are shown for six-test electrons chosen from all of the accelerated electrons. The time history of orbits x(t) and momentum p_x, is plotted in Figures 7 and 8, respectively. The scenarios shown in these two figures are consistent with our discussions above.

Time history of orbits x(t) for six representative test particles, in the case of laser pulse, a = 1.0.

Time history of longitudinal momentum px for six-test electrons, for laser pulse, a = 1.0.

When intense laser propagates in the low-density plasma, a large amplitude plasma wave is generated mainly by stimulated Raman scattering (Mima et al., 2001). Rapid SRS heating in forward direction produces a massive electron (cloud) blow-off into the vacuum region. The large electrostatic field builds up to drive return currents and accelerate the background electrons to high energies. As shown in Figure 4, for the electrostatic field E_x plots, at time t = 196.99 (the laser pulse has passed through the right boundary) SRS driven large plasma wave is propagating. At later times, as SRS gradually saturates, an intense narrow ES field structure (Debye sheath) with a steep gradient forms at each plasma interface. The maximum electrostatic field goes up to E_x^max ≈ 0.18mω₀ c/e. The electrons get accelerated backward by this sheath electric field, which spreads over the hot-electron Debye length (Mackinnon et al., 2002). Assuming that the electron with maximum energy is accelerated by the gradient field beyond the right boundary; the acceleration distance (Fig. 4, t = 321.46) roughly equals the plasma length L = 1000c/ω₀. Electron experiences a linearly-increasing electrostatic field E_x(x) ∼ 0.0001x; the maximum gained electron energy is approximately, E^max ≈ ∫₀^L eE_x(x) dx ≈ 50 MeV which well agrees with the result E^max ≈ 46 MeV, measured above. This leads to impressive effective accelerating electric fields of the order of hundreds of GV/m.

In our simulations, plasma density is taken as n = 0.01n_cr, that is, the ratio of the laser to the plasma frequency is ω₀ /ω_p = 10, that is, the plasma wave period T_p = 10T₀, where T₀ is laser pulse period. As known, due to large ion inertia M_i = 1836m_e, the ion plasma frequency is only about ω_ion = (m_e /M_i)^1/2ω_p ≈ ω_p /43, the ion plasma wave period T_ion ≈ 43T_p = 430T₀, 430 times the laser period, the total simulation time is about t ≈ 1300T₀ ≈ 3T_ion. Therefore, it seems plausible to neglect the ion dynamics, especially in the formation of the short high-quality, well-collimated return beam of relativistic electrons. Our additional PIC runs have proved that above assertion is correct.