2.1.1. Hamiltonian formulation of the problem, integrability of the system

Let us consider a charged particle in an electromagnetic plane wave propagating along the z direction (the wave vector k₀ is parallel to the z direction). The following four-potential is chosen:

where E₀, ω₀, and k₀ are constants.

When time t is treated as a parameter entirely distinct from the spatial coordinates, the relativistic Hamiltonian of a particle in the wave is

where −e, m, and P_i(i = x, y, z) are, respectively, the particle's charge, its rest mass, and its canonical momentum components. This system has three degrees of freedom. We next introduce the following dimensionless variables and parameter

and perform the canonical transformation,

, given by the type-2 generating function:

. This canonical transformation keeps

unchanged and yields:

. The new Hamiltonian expressed in terms of the new conjugate variables, is:

(Lichtenberg & Liebermann, 1983). This Hamiltonian,

are three constants of motion, which are independent and in involution. As a consequence, the system is completely integrable (Lichtenberg & Liebermann, 1983).

2.1.2. Integration of the Hamilton-Jacobin equation

When using the proper time of the particle to parameterize the motion in the extended phase space, the Hamiltonian of the charged particle in the wave reads (Jackson, 1975): H₀ = (1/2)mc²γ² − (1/2m)(P + eA)² − (1/2)mc². When the dimensionless variables and parameter (Eq. 2) are used again, the normalized Hamiltonian is:

. A normalized proper time:

is also introduced.

Although the electron motion is not restricted to the plane of polarization of the wave (the y degree of freedom is assumed to be excited), we look for a set of actions (P_⊥,P_//,E) and angles (θ,φ,φ), instead of the configuration (r,t), and momentum, (P,−γ) in the

phase space. This comes out to say that we seek a canonical transformation

, such that the new moments are constants of motion. Following Landau and Lifshitz (1975), the following type-2 generating function is obtained (Rax, 1992)

The Hamiltonian in terms of the action variables is:

where M² = 1 + a²/2 (Rax, 1992). As

, the energy momentum dispersion relation is given by the following expression:

.

The solution of Hamilton-Jacobin equation is

This solution is necessary to predict resonances when a perturbing mode is considered.

2.2.1. The perturbing waves have their electric fields perpendicular to the polarization plane of the high intensity wave

Let us consider a transverse perturbation polarized perpendicularly to the plane of polarization of the high intensity wave. The wave vector of the perturbing mode with vector potential A₁ is assumed to be at some angle α with respect to the wave vector of the high intensity wave, one has:

In the extended phase space, when using the proper time of the particle to parameterize the motion, the Hamiltonian of the particle can be put in the following form (Jackson, 1975; Rax, 1992)

or, neglecting terms in A₁²:

where H₀ is the Hamiltonian of the system when the perturbation, H₁, is neglected. We know that this problem is integrable. We chose:

.

The dimensionless variable

is added to the dimensionless variables and parameter defined by equations (2). They are used to normalize the equations of motion. A normalized vector potential for the perturbation A₁, a₁ = eA₁ /mc = eE₁ /mcω₁, and a normalized proper time,

, are also introduced. The normalized Hamiltonian is given by:

, the normalized perturbing vector potential is:

, with

. We have assumed that

, which means that we consider that

.

First, the phase,

, is expressed in terms of the coordinates defined by the transformation generated by (4). We have: κ = κ′ − δ sin ζ − β sin 2ζ, with:

. Then, one obtains

The generalized Bessel function which is usually defined as

is introduced (Nikishov & Ritus, 1964). Letting N = (n + 2m), one obtains the Hamiltonian expressed as a sum of harmonic interactions

with V_N(δ,β) = (−1)^NC_−N(δ,β). When the zero-order solution (4) is plugged in the argument of the perturbation sines, the stationary phase condition leads to the following resonance condition (Rax, 1992)

This resonance condition when restricted to the energy surface equation given by the following expression:

gives P_⊥ versus P_//. We have plotted these resonances in the case when

in (9). Considering that

comes out to assuming a higher value of

, given by

. When

, Figure 1 shows P_⊥ versus P_// when the resonance condition is satisfied, for α = π/4 and α = 3π/4. The solid lines are obtained when assuming that the perturbation is such that

. The dashed lines correspond to the situation when

. Figure 1 shows that the lines are quite far from each other when considering one perturbing wave only. One can conclude that a second perturbing wave is necessary so that many resonances overlap. As a consequence, two perturbing waves were considered in order to define conditions when efficient stochastic heating takes place. We consider that the two wave-vectors of the perturbation are symmetric with respect to the direction of propagation of the very intense wave.

Resonances in the (P//,P⊥) plane, . (a): α = π/4, (b): α = 3π/4.

The whole numerical results obtained show that the resonance overlap seems to be easier (as the resonances get closer to each other) when

go to higher values.

In order to calculate the width of the N^th resonance, following Rax (1992), a resonant torus (P_//c, P_⊥c, E_c) verifying the resonance condition Eq. (9) for a certain value of N is isolated. The following new variables are introduced next:

Then, the perturbed motion is described by a one-dimensional oscillator. The half-width of this resonance is given by:

(Lichtenberg & Liebermann, 1983). Consequently, the half-width in terms of the action variables is given by

. Then, the sum of the half-widths of the two resonances N and N′, corresponding to modes with wave vectors

, can be calculated. The following quantities: ΔP_N⊥ + ΔP_N′⊥ and ΔP_N// + ΔP_N′// can be estimated in order to apply Chirikov criterion.

Let us consider now, the overlap of resonances N = −1 corresponding to the two modes with wave vector components

when

. The quantities V₋₁,k_⊥ and V₋₁,−k_⊥ are evaluated by using the following Taylor expansion relevant for small δ and β (Rax, 1992; Ritus, 1985): C₁(δ,β) = δ/2 + δβ/4 − δ³/16. In the situations considered here, the results obtained when using this expansion are in good agreement with those obtained through a numerical evaluation of the series representation of this generalized Bessel function. The Chirikov (1979) criterion is satisfied for two resonances when their unperturbed separatices touch or overlap. In this case, a trajectory is no longer locked within one of the resonance, and it can pass from one resonance to the other. In other words the Chirikov threshold is reached when ΔP_−1⊥ + ΔP_−1⊥′ and is ΔP_−1// + ΔP_−1//′ are larger than the distance between the two resonances in term of P_⊥ and P_//, respectively. The values of the quantities and which are used in our numerical estimates verify the resonance condition (9).

When assuming that

, the resonance overlap seems to be easier to achieve. Then, the most dangerous resonance is N = −3. In order to calculate the resonance half-widths of each resonance, the generalized Bessel function, C₃(δ,β), is estimated through its series representation.

When

, considering P_// = 0.4 and P_⊥ = ±0.231, we are well above the Chirikov threshold for the resonance N = −3. Figure 2 shows the evolution of the energy of one particle in the case when it interacts with the three modes and in the case when all the intensity is in one mode. This figure shows that there is stochastic heating for realistic laser parameters.

Normalized energy of the charged particle versus time. . (α) a = 2, a1 = 0.2, (β) the total intensity is in one wave: a = 2.176, a1= 0.

2.2.2. The perturbing waves have their electric fields in the polarization plane of the high intensity wave

The normalized vector potential is assumed to be

The perturbing Hamiltonian is still approximated by:

.

The same calculations as in the previous chapter were performed. The same resonance condition was obtained (Eq. 9). Stochastic heating was also evidenced considering one particle only.

2.2.3. PIC code simulation results

PIC code simulations were performed in order to confirm stochastic heating. As we used the CALDER code in its two-dimension version, only the case when the electric fields of the perturbation are in the plane of polarization of the high intensity wave was considered (Lefebvre et al., 2003; Pommier & Lefebvre, 2003). The simulation box is 9.55 × 15.9 μm² with homogeneous plasma. Laser parameters typical of current terawatt facilities were considered to study the influence of the angle

. The wavelength is λ₀ = 1 μm, the length of the pulse is Δτ = 0.3 ps, and the maximum intensity is: I = 5.53 × 10¹⁸ W/cm². The initial plasma temperature was assumed to be 1 keV.

First, stochastic heating was evidenced considering the energy carried by all the waves is the same in all cases. The case when the plasma interacts with three waves was compared to the one when it interacts with one wave only. Both electron temperature and total kinetic energy are higher in the cases with three waves (Fig. 3a). As the ponderomotive force is almost the same in the two situations, one can conclude that this is due to stochastic heating. The effect of stochastic heating will appear more clearly when considering two counter-propagating long pulses.

Evolution of the plasma kinetic energy. (a): ne = 0.01nc; (1): a =2, a1 = 0.1, α = 5π/6; (2): a = 2, a1 = 0.1, α = π/6; (3): one wave case, a = 2.002. (b): ne = 0.05nc; a = 2, a1 = 0.1; (1): α = 5π/6; (2): α = π/6.

Figure 3a and 3b show that stochastic heating grows with α and the plasma density. As a consequence, the best way to obtain efficient stochastic heating is to consider two counter-propagating waves. To study this specific problem, one-dimensional (1D) simulations were achieved. The laser pulses are two stepwise 1.59 ps long pulses with the same frequency. The plasma occupies a 100 μm region, the laser wave length is 1 μm and the maximum intensity for the high intensity wave is I = 1.24 × 10¹⁹ W/cm² (a = 3), and for the low intensity wave I₁ = 1.38 × 10¹⁶ W/cm² (a₁ = 0.1). The initial temperature of the plasma is still 1 keV. The energy carried by all the waves was kept constant. The case when the plasma interacts with three waves is compared to the one when it interacts with one wave only (Fig. 4). The difference between the one wave case and the two wave's case shows that stochastic heating can be very important when considering a high intensity long pulse. These results show that stochastic heating can be predicted with a plane wave approach.

Evolution of the plasma kinetic energy in the case of one wave and two counter-propagating waves. a = 3, a1 = 0.1, ne = 0.01nc.