2.1. Hamiltonian formulation of the system

A charged particle interacting with two electromagnetic planes waves is considered. One has a high intensity:

and the second, perturbing wave, a₁, is in the same polarization plane propagating at some angle α with respect to a₀ (a₁ << a₀):

with k_∥ = k₁ cos α and k_⊥ = k₁ sin α, where

. In the following we will use e = m = c = 1.

The Hamiltonian of an electron interacting with both the

waves is:

We rewrite H as H = H₀ + H₁, where H₀ is the integrable part of the Hamiltonian and H₁ is the perturbation. In order to write H in the action-angle variables of H₀, we used the relations (Bourdier et al., 2005; Bourdier & Patin, 2005; Patin et al., 2005; Rax, 1992):

where p_x (resp. p_z) is the normalized momentum along the x-axes (resp. z-axes), and H₁ expressed in terms of the action angle variables (P_⊥,P_∥,E,θ,φ,φ) of H₀ (Rax, 1992).

We can notice that P_⊥ = 〈p_x〉, P_∥ = 〈p_z〉 and E = 〈γ〉.

One has:

Neglecting the a₁² term:

where V_N = V_N(P_⊥,P_∥,E) is the amplitude of the Nth resonant term.

2.2. Resonance condition and stochasticity threshold

The resonance condition is found by using the standard perturbation technics (Tajima et al., 2001; Patin et al., 2005; Rax, 1992):

As H₀ = 0, one has:

Using this equation and Eq. (12) allows to calculate P_⊥ versus P_∥. Figure 1 displays the resonance lines for α = π/6, and shows that these lines are quite far from each other in this case. Chaos will occur when the sum of th half-widths of two neighboring resonances, ΔJ, becomes larger than the distance between them, d. This condition is known as the Chirikov criterion (Chirikov, 1979):

Resonance in the (P∥,P⊥) plane, a0 = 2, ω1 = k1 = 1 and α = π/6.

We will show below that the resonance width is weakly dependant on the angle α. Therefore, we expect that chaos will set in easily if the resonance lines are closer, as is the case in Figure 2 for α = 5π/6. In order to prove this assertion rigorously, we need to calculate the width of the resonance. Following Rax (1992) and Tabor (1989), the resonance width is given by:

We note that ΔJ depends on α through its sine and cosine (remembering that k_∥ = k₁ cos α and k_⊥ = k₁ sin α). So, the width of the resonance is not strongly dependent on α. Remembering that the aim is to have global stochasticity, we must find conditions such that the Chirikov criterion is satisfied as widely as possible.

Resonance in the (P∥,P⊥) plane, a0 = 2, ω1 = k1 = 1 and α = 5π/6.

First, we compute the Chirikov criterion parameter in the {α;ω₁} space. Figures 3 and 4 display the region in the {α;ω₁} space where R > 1. In other words, it is the region where the Chirikov criterion is satisfied and stochastic heating should be strong. Figures 3 and 4 are symmetric with respect to the α = π axis. Furthermore, the Chirikov criterion is satisfied when α is close to π and when ω₁ is in the range (Tajima et al., 2001; Bourdier et al., 2005).

Zone (view 1) (in the (α;ω1) space) where the Chirikov criterion is satisfied for resonances N = −1; N = −2 with P⊥ = 0; a0 = 1 and a1 = 0.1.

Zone (view 2) (in the (α;ω1) space) where the Chirikov criterion is satisfied for resonances N = −1; N = −2 with P⊥ = 0; a0 = 1 and a1 = 0.1.

It is also interesting to know the wave-intensity threshold for the stochastic effect. We can know this by plotting the Chirikov criterion parameter in the {a₀;a₁} space, for given values of α and ω₁. We can see clearly in Figures 5 and 6 that there is a threshold in a₁ for the stochastic heating effect.

Zone (view 1) (in the (a0;a1) space) where the Chirikov criterion is satisfied for resonances N = −1; N = −2 with P⊥ = 0; ω1 = 1 and α = 5π/6.

Zone (view 2) (in the (a0;a1) space) where the Chirikov criterion is satisfied for resonances N = −1; N = −2 with P⊥ = 0; ω1 = 1 and α = 5π/6.

2.3. Numerical resolution of the theoretical model's equation

Several numerical results are presented in order to validate the theoretical hypothesis. First,we compare the particle energy in two different cases corresponding to an one-wave case, and a two-wave case. Figure 7 shows the particle energy versus the time calculated through the Hamiltonian one-particle model in the one-wave case. Figure 8 is for the two-wave case. In this last case, the particle has a chaotic motion, its average energy is higher due to the stochastic heating. It is also interesting to plot the behavior in the (p_x; p_z) space. Figures 9 and 10 display the motion in the (p_x; p_z) space. In the one-wave case (Fig. 9), the particle is in one resonance, whereas in Figure 10, the particle can travel from one resonance to another. Remembering that P_⊥ = 〈p_x〉, P_∥ = 〈p_z〉, we notice that the particle fills the resonance diagram (Fig. 2). Now, we can highlight the stochastic threshold in intensity by computing the motion in the (p_x; p_z) space for two different values of a₁.

Hamiltonian versus time in the one-wave case with a0 = 4.02.

Hamiltonian versus time in the two-wave case with α = 5π/6, ω1 = k1 = 1, a0 = 4.0 and a1 = 0.4.

Motion in the (px; pz) space in the one-wave case with a0 = 4.02.

Motion in the (px; pz) space in the two-wave case with α = 5π/6, ω1 = k1 = 1, a0 = 4.0 and a1 = 0.4.

Figures 11, 12, 13, and 14 are all for two-waves cases. Figures 11 and 13 are for a₁ = 0.2. Figures 12 and 14 are for a₁ = 0.3. The difference is striking, with a very regular motion for the two-wave case at low perturbing wave intensity, and the expected chaotic behavior at larger value of a₁. It is explained by Figure 6. Indeed, in one case (a₁ = 0.2), the Chirikov criterion is not satisfied and, in the other case (a₁ = 0.3), the Chirikov criterion is satisfied. In consequence, the particle is free to pass from resonance to resonance and can reach higher energies.

Motion in the (px; pz) for a0 = 4, ω1 = 1, α = 5π/6 and a1 = 0.2.

Motion in the (px; pz) for a0 = 4, ω1 = 1, α = 5π/6 and a1 = 0.3.

Hamiltonian of the particle versus time for a0 = 4, ω1 = 1, α = 5π/6 and a1 = 0.2.

Hamiltonian of the particle versus time for a0 = 4, ω1 = 1, α = 5π/6 and a1 = 0.3.

3.1. PIC simulation with one particle

Before addressing the interplay between collective and individual absorption mechanisms, we first wanted to make sure that the stochastic behavior was correctly modeled by the PIC code. We therefore started with simulations including a very-low-density plasma, with only one particle. Figure 15 displays the particle energy versus time in two cases. The black curve correspond to the one-wave case. As it is well known, the energy oscillate in the electromagnetic wave. The red curve is for the two-wave case. In this case the energy curve has a chaotic behavior.

Kinetic energy as a function of time computed with CALDER for two sets of parameters (see text).

The Figures 8 and 15 are of the same type. The differences come from the fact that in the PIC code simulation we have a half-plane wave. So there is a discontinuity in the electromagnetic field; the particle undergoes a kick. The system is chaotic, as a consequence, a minor change in the initial conditions or in the value of the electromagnetic field changes the trajectories. Furthermore, the particle motion integrators in the PIC code and for the numerical resolution of Hamilton's equation are different. So it is not surprising that the details of the chaotic curves are different in Figures 8 and 15. Figure 16 displays the particle trajectory in the (p_x; p_z) phase space computed with the PIC code. Again, the motion is absolutely similar to the one obtained in Figure 12.

Motion in the (px; pz) using the code CALDER with a0 = 2, ω1 = 1, α = π and a1 = 0.1.

3.2. Influence of the angle between the waves

To assess the relevance of stochastic heating in realistic situations, we performed a series of two-dimensional-simulation, where we have a₀ = 1 (amplitude of the high intensity wave), a₁ = 0.1 (amplitude of the perturbing wave), τ₀ = 0.60 ps (last of the high intensity wave), τ₁ = 0.62 ps (last of the perturbing wave), T_e,i = 1 keV (initial temperature of the electrons), and n = 0.01 n_c (density of the plasma). Figure 17 displays the evolution of the kinetic energy of the electrons when α = 5π/6, α = π/6, and α = 0. It shows that the highest energy transferred to the electrons is when α = 5π/6.

Kinetic energy versus time in three cases: α = 5π/6: red curve; α = π: blue curve; one wave case: green curve.

The energy of the electrons in the one-wave case is due to the ponderomotive force. In the other two cases, the ponderomotive effect still exists, but a new phenomenon takes place due to the stochastic heating effect. We can conclude two things. First, the stochastic heating mechanism can be readily observed with PIC code simulations. Then, the stochastic heating is shown to be more efficient when the waves are almost, but not exactly, counter propagating. According to the theoretical model, the Chirikov criterion is indeed achieved more easily when α = 5π/6.

3.3. Influence of ω₁

In the next series of two-dimensional-simulations, the parameters are a₀ = 3.922, a₁ = 0.784, n = 10⁻²n_c, τ₀ = τ₁ = 0.3 ps, and α = 5π/6. Figures 18 and 19 show that the gain is inversely proportional to ω₁. Indeed, Figures 3 and 4 show that the Chirikov criterion is less satisfied when ω₁ is greater. Furthermore, Eq. (15) is inversely proportional with respect to ω₁.

Kinetic energy of the plasma.

Electron energy distribution.

3.4. Influence of q = a₀ /a₁ for a given laser energy

The next simulations was performed with three waves, one with a high intensity and two perturbating waves symmetric with respect to propagation axis of the intense one. The physical parameters of the two-dimensional-simulations are

, where n is the density of the plasma, τ₀ and τ₁ are the length of the two pulses and T_e is the initial temperature of the electrons. The gain is defined by:

where Ek_{3 waves, f} (resp. Ek_{1 wave, f} ) is the kinetic energy of the particles at the end of the simulation for the three wave case (resp. one wave case). The kinetic energy of the electrons is compared to the case when there is only one wave with the same laser energy (i.e., with an a equal to

). The relative gain and the absolute gain reach a maximum when the two counterpropagating waves amplitude ratio is 50% (Table 1). In this case, absorption is increased by more than one order of magnitude compared to the one-wave case.

Gain for different value of a0 /a1. We have Ek1 wave, f = 8.23·10−4 (a.u.)

3.5. Influence of the density

The simulation parameters used in the simulation are a₀ = 3.922, a₁ = 0.888 (two perturbating waves), and T_e = 1 keV. The density is varied from 10⁻⁵n_c to 10⁻¹n_c. The relative gain is optimum for n = 5·10⁻²n_c (cf. Table 2), but the absolute laser absorption is highest for n = 10⁻¹ n_c in the three waves case. It is interesting to note that the energy deposited in the plasma is larger in the three waves case at n = 5·10⁻² n_c than in the one wave case at n = 10⁻¹ n_c.

Gain for a0 = 3.922