SIMULATION MODEL AND RESULTS

To study the generation of the hemispherical fast electron waves, the relativistic 2D3V particle-in-cell (PIC) code LAPINE (Xu et al., Reference Xu, Chang and Zhuo2002) is used to simulate the interaction of an ultraintense laser pulse with the target. A plasma slab with density 50n _c and thickness 5λ₀ is preceded by a subcritical preplasma, where n _c = 1.12 × 10²¹ cm⁻³ is the critical density and λ₀ = 1 µm is the laser wavelength. The density of the latter increases exponentially from 0.02n _c to n _c with a scale length L _p = 8 µm. In order to preclude the effects of ion acceleration and heating (Pretzler et al., Reference Pretzler, Saemann, Pukhov, Rudolph, Schätz, Schramm, Thirolf, Habs, Eidmann, Tsakiris, Meyer-ter-vehn and Witte1998), the plasma is assumed to consist of electrons and Al ions with mass 27m _p and charge 10e, where m _p = 1836 m _e is the proton mass. The initial temperature of both the electrons and ions is 1 KeV. The simulation box is 60λ₀ × 30λ₀ with 2400 × 1200 cells, and 49 electrons and 16 Al ions per cell. A 10²⁰ W/cm² (a ₀ = 8.54) p-polarized laser pulse is incident normally from the left. Both the spatial and temporal profiles of the laser pulse are Gaussian. The laser spot radius is 5λ₀ and the pulse duration is 40T ₀ (FWHM), where T ₀~3.3 fs is the laser period. The simulation time step is 0.009T ₀. Absorbing boundary conditions are used for the fields and particles at all boundaries.

Figure 1 shows the distributions of the electron density (a), kinetic energy density (b), kinetic energy density along the radial direction (c), and the electron energy spectrum (d). One can see that a plasma channel is formed in the preplasma and several nonlinear processes take place there, including self-focusing, hole-boring, as well as filamentation, all of which can affect the generation and propagation of the fast electrons. The electrons are accelerated by the laser pulse as it propagates in the preplasma. The energetic electrons then penetrate the solid-density slab and enter into the backside vacuum region, where they exhibit a periodic hemispherical distribution with ~0.2n _c density peaks and laser wavelength spacings, as shown in Figure 1c, in which the electron kinetic energy density distribution along the radial direction in the upper half-space is presented. Sun et al. (Reference Sun, Ott, Lee and Guzdar1987) showed that a laser pulse propagating in underdense plasma with a frequency of ω_p experiences relativistic self-focusing if the laser power P exceeds the critical power P _cr ≈ 17(ω₀/ω_p)² GW, where ω₀ is the laser frequency. For the average density (~0.25n _c) of the preplasma, we obtain P _cr ~ 6.7 × 10¹¹ W (I = 2.7 × 10¹⁷ W/cm² for the 5 µm spot radius here), so that the self-focusing condition is satisfied. Figure 2 shows the Poynting vector along the laser propagation axis. It can be seen that the laser pulse is indeed self-focused in the preplasma, resulting in a tight focal spot (~2 µm) at the solid-slab front. The phase velocity of the wave fronts propagating through a focusing medium can be approximated as (Gibbon, Reference Gibbon2005)

$$\eqalign{v_p \lpar r\rpar & =c\mathop {\left[{1 - \displaystyle{{{\rm \omega} _p^2 } \over {{\rm \omega} _0^2 \lpar 1+a^2 \lpar r\rpar /2\rpar ^{1/2} }}} \right]}\nolimits^{ - 1/2} \cr & \sim c\left[{1+\displaystyle{{{\rm \omega} _p^2 } \over {2{\rm \omega} _0^2 }}\left({1 - \displaystyle{{a^2 \lpar r\rpar } \over 4}} \right)} \right]\comma}$$

where c is the light speed in vacuum and a(r) is dimensionless radial profile of the laser. It shows that the phase fronts will propagate slower at the center than at the edge, thus inducing a curvature in the phase front and causing the light to bend. We can indeed see that self-focusing bends the laser wave fronts to a converging hemispherical profile. Figure 1b shows that the fast electron layers in the upper (lower) vacuum regions behind the target are generated in the lower (upper) regions of the preplasma. That is, the fast electrons from the preplasma cross the midplane as they propagate through the solid target layer. This is because the laser-light polarization in the preplasma is locally modulated by the self-focusing, leading to a v × B force on the electrons that drives them toward the midplane and beyond, while the electrostatic space-charge field arising from the laser expulsion of the preplasma electrons confines the latter's transverse excursion. The electron temperature can reach 7.8 MeV, with a cut-off energy at ~65 MeV, as can be seen in Figure 1d. The temperature T _h of fast electrons generated in intense laser-plasma interaction as given by the Wilks scaling law $T_h=m_e c^2 \lpar \sqrt {1+a_0^2 } - 1\rpar $ (Wilks et al., Reference Wilks, Kruer, Tabak and Langdon1992) is T _h = 3.9 MeV (for a ₀ = 8.54 as here), which is about half of that obtained in our simulation. Thus, the preplasma can enhance the acceleration of electrons. We also noted that the temperature distribution of the fast electrons behind the solid-slab is similar to that in Figure 1d except that the electron number is reduced slightly.

Fig. 1.

(Color online) log₁₀ of electron density (a) and electron kinetic energy density (b) distribution at t = 90T ₀, respectively. The distribution of electron kinetic energy density along the radial direction at t = 90T ₀ (c), whose origin of coordinate is at the midplane of the solid-slab front. The energy spectrum of the electrons at t = 100T ₀ (d). The density is in units of n _c and the kinetic energy density is in units of m _ec ²n _c (same in the other figures).

Fig. 2.

(Color online) log₁₀ of Poynting vector along the laser propagation axis (${\displaystyle{c \over {4{\rm \pi} }}}\lpar E \times B\rpar _z $) at t = 70T ₀, only the positive energy flux is shown. The energy flux is in units of m _e²c ³ω₀³/4πe ².

Several mechanisms can produce high energy electrons in the underdense plasma. These include betatron resonance absorption, stochastic heating, and LWFA. In the present problem, the laser pulse duration is much larger than the plasma wavelength and the acceleration distance is very limited, so that the contribution of LWFA can be neglected. Stochastic heating can also be neglected. Figures 3a and 3b for the transverse and longitudinal electron phase space shows that the electrons oscillate in the laser electric field at the laser frequency. The transverse momentum of the electrons increases with the laser penetration distance and can reach a maximum of ~70m _ec, which is about eight times higher than that (namely P _y = a ₀m _ec = 8.5m _ec) of a relativistic electron in a plane electromagnetic wave in vacuum (Yu et al., Reference Yu, Bychenkov, Sentoku, Yu, Sheng and Mima2000). Since the electrons are also accelerated by the reflected laser pulse propagating backward in the preplasma, as indicated by the negative electron momentum (P _z < 0) near the slab target in Figure 3b, their transverse momentum is further increased. The transverse electron momentum in the wave is converted into longitudinal momentum by the v × B interaction, and P _z of the fast electrons can reach ~110m _ec, about three times that, namely $P_z={\textstyle{1 \over 2}}a_0^2\, m_e c=36.5m_e c$ in vacuum (Yu et al., Reference Yu, Bychenkov, Sentoku, Yu, Sheng and Mima2000). As expected, we see that the oscillation frequency of the longitudinal momentum is twice that (ω₀) of the transverse momentum, accompanied by significant longitudinal heating (spread in the z − P _z phase space).

Fig. 3.

(Color online) Distributions of the electron (a) transverse momentum (P _y) and (b) longitudinal momentum (P _z), and the (c) transverse quasistatic magnetic field (B _x), and (d) transverse electrostatic field (E _y) at t = 80T ₀. The momentums are on log₁₀ scale. The fields B _x and E _y have been averaged over eight laser periods and are in units of m _ecω₀/e.

The ponderomotive expulsion of electrons produces strong transverse charge-separation fields, as well as electron currents that generate quasistatic magnetic fields (Pukhov et al., Reference Pukhov, Sheng and Meyer-ter-vehn1999), as can be seen in Figures 3c and 3d, where for clarity these fields have been averaged over eight laser periods. We can see that the quasistatic fields E _y and B _x remain periodic in space (whose spatial periods are comparable to the laser wavelength), and their maximum amplitudes occur at the boundaries of the self-focusing cone, where the electron density is also the highest. Accordingly, electrons are accelerated forward in layers into the opposite half-space behind the target by the v _yB _L force, where B _L is laser magnetic field and v _y is the speed of the electrons in the charge-separation field E _y. This also explains why the hemispherical electron layers have a slight phase mismatch at the midplane.

In order to see how the hemispherical electron waves are generated, we have also considered the interaction of a laser pulse with a solid density target without a preplasma, as well as with only a preplasma. Figure 4a shows that no hemispherical electron waves appear when there is only the solid-slab. As expected, only electron clouds appear at the front and back of the latter (Wilks et al., Reference Wilks, Kruer, Tabak and Langdon1992). On the other hand, Figure 4b shows that with only the preplasma, hemispherical waves are generated and propagate forward into the target backside region. That is, the preplasma is both necessary and sufficient for the generation of these waves. The temperature of the fast electrons can reach ~6.3 MeV, which is lower than that for the complete preplasma-solid layer system (Fig. 1). This can be attributed to the absence of laser pulse reflection, so that the recoil acceleration of electrons as well as additional heating by the reflected wave do not occur. In fact, most of the laser pulse penetrates through the target, with only a small amount of reflection from the conic regions where the density is snow-plowed to above-critical by the ponderomotive compression and self-focusing. From the frequency spectra of the self-generated electric fields for all the cases considered, we could not find evidence of stimulated Raman scattering (Gibbon, Reference Gibbon2005). On the other hand, the energetic electrons oscillate periodically in both the transverse and longitudinal directions in the fields of the incident and reflected light waves, which means that the contribution of the stochastic heating by counter-propagating electromagnetic waves is insignificant in the situation. For completeness, Figures 4c and 4d show the electron kinetic energy density distributions for interactions of a circularly polarized (CP) and s-polarized (SP) laser pulse with the target plasmas. One can see that fast hemispherical electron waves with density peaks of ~0.1n _c are generated by the CP pulse, and the crossed propagation of the hemispherical waves in the target, as discussed above. Although the 2ω₀ component from the J × B heating (Kruer, Reference Kruer1985) is nearly absent, the electron temperature can still reach 5.2 MeV. However, hemispherical fast electron waves do not appear for the SP pulse. This can be attributed to the fact that here the laser electric field is not in the simulation plane. The electrons can still oscillate in the laser electric field, but the corresponding force of the quasistatic fields is missing in the electrons oscillating plane due to our 2D3V simulation. That is, the betatron resonance of the electrons does not occur.

Fig. 4.

(Color online) log₁₀ of electron kinetic energy density distributions for the cases with a solid target without the preplasma (a), target of preplasma only (b), circularly polarized laser (c), and s-polarized laser (d) at t = 90T ₀. The other parameters are the same as that in Figure 1.

For completeness, we also considered the effect of the scale length of the preplasma on the generation of hemispherical fast electron waves. One can see in Figure 5a that fast electron bunches with half wavelength spacing can be generated in a preplasma with L _p = 2 µm, but they do not have an obviously hemispherical profile. In addition, self-focusing of the laser pulse in the preplasma and the crossed propagation of the fast electron bunches in the target are not observed. In this regime, the J × B heating is still dominant, which leads to an electron temperature of 3.1 MeV, consistently with the Wilks' scaling (3.9 MeV). With increasing preplasma scale length, the hemispherical wave profile becomes more and more obvious. In fact, for L _p = 12 µm we still find fast hemispherical electron layers with ~0.2n _c density peaks, as shown in Figure 5b, very similar to that for L _p = 8 µm. Because of limitation of our computation sources, we have not considered still longer scale lengths. However, it can be expected that hemispherical waves can still be generated as long as the laser pulse self focuses appropriately.

Fig. 5.

(Color online) log₁₀ of electron kinetic energy density distributions for the cases with scale length of preplasma 2 µm at t = 70T ₀ (a) and 12 µm at t = 120T ₀ (b), respectively. The laser intensity is fixed at 10²⁰ W/cm².

The effect of laser intensity on the generation of the hemispherical fast electron waves has also been investigated. In Figure 6, we show the electron kinetic energy density distribution for the cases with laser intensity 10¹⁸ W/cm² and 10²¹ W/cm². It is found that the amplitude of the waves behind the target increases with the laser intensity. Furthermore, no waves are generated behind that target for a = 10¹⁸ W/cm². This is consistent with the observation (Pukhov et al., Reference Pukhov, Sheng and Meyer-ter-vehn1999) that for betatron resonance the laser power should significantly exceed (e.g., six times) the critical power for self-focusing.

Fig. 6.

(Color online) log₁₀ of electron kinetic energy density distributions for the cases with laser intensity 10¹⁸ W/cm² (a) and 10²¹ W/cm² (b) at t = 90T ₀, respectively. The scale length of the preplasma is fixed to 8 µm.

In Figure 7a we show the dependence of the fast-electron temperature T _f on the preplasma scale length and the laser intensity. We see that the fast-electron temperature first increases rapidly with the preplasma scale length and then the increase slows down, although saturation was not found because of limitation in the size of our system. Figure 7b shows the dependence of the fast-electron temperature on the laser intensity for L _p = 8 µm. As expected, we have T _f ~ 0.87a ₀, or T _f ~ 0.74(I ₁₈λ_μm²)^1/2, where I ₁₈ is the laser intensity in units of 10¹⁸ W/cm² and λ is the laser wavelength in units of  µm. The fast-electron temperature obtained here is about half that in Pukhov et al. (Reference Pukhov, Sheng and Meyer-ter-vehn1999), where a much longer preplasma (L _p = 30 µm) was used.

Fig. 7.

(Color online) Fast electron temperature as a function of preplasma scale length (a) and dimensionless laser electric field (b), respectively. The laser intensity is fixed at 10²⁰ W/cm² in (a), and the scale length of the preplasma is fixed to 8 µm in (b). The dots are PIC simulation results and the solid lines are the best fits of the latter.