Stochastic heating of free electrons in multiple electromagnetic waves: A simple physical analysis

It is established that charged particles crossing the interference field of two colliding electromagnetic (EM) waves can behave chaotically, leading to a stochastic heating of the particle distribution. A fine understanding of the stochastic heating process is crucial to the optimization of many physical applications requiring a high EM energy deposition to these charged particles. Predicting key stochastic heating features (particle distribution, chaos threshold) is usually achieved using a heavy Hamiltonian formalism required to model particle dynamics in chaotic regimes. Here, we explore an alternative and more intuitive path, which makes it possible to reduce the equations of motion of particles to rather simple and well-known physical systems such as Kapitza and gravity pendulums. Starting from these simple systems, we first show how to estimate chaos thresholds by deriving a model of the stretching and folding dynamics of the pendulum bob in phase space. Based on this first model, we then derive a random walk model for particle dynamics above the chaos threshold, which can predict major features of stochastic heating for any EM polarization and angle ${\theta }_i$.

Figure 1

Electron dynamics in a single linearly polarized EM wave. In panel (a), a snapshot at a given time $t$ of the laser spatial profile is pictured in color scale. The laser is traveling towards positive $x$ through a cloud of electrons (considered as noninteracting test particles) pictured as black dots. One particle is highlighted as a blue dot (in all panels), and its trajectory up to time $t$ is shown as a blue line. In panels (b) and (c) we also display a snapshot, at the same time $t$, of the electron distributions in ($x,p_x$) and ($p_x,p_z$) spaces, respectively. To assist the reader, the magnetic field is plotted as well in panel (b) (red line).

Figure 2

Formal analogy between a pendulum and the dynamics of an electron in two colliding waves. Each panel presents a different interaction configuration: (a) interaction of an electron with only one linearly polarized wave, (b) with two RHC circularly polarized waves, (c) with one RHC- and one LHC-polarized wave, and (d) with two linearly polarized waves. For each case, we remind the reader of its equivalent pendulum: (a) and (c) simple gravity pendulum, (b) pendulum at rest, and (d) Kapitza pendulum.

Figure 3

Electron dynamics in RHC/LHC configuration. Panel (a) shows the field resulting from the interference of counterpropagating RHC-polarized and LHC-polarized waves. Each line style corresponds to a different time within a laser period. In addition, the positions of magnetic field nodes are highlighted by vertical dashed lines. In panels (b) and (c), we represent the electron distributions in phase space after $2T_0$ in the fields of panel (a) for $a_0=0.8$, when using the nonrelativistic (b) and relativistic (c) forms of the equations of motion. The discrete color scale (right) encodes the different buckets in which particles were initially located.

Figure 4

Electron dynamics in L/L configuration. Similar to Fig. 3 but for two linearly polarized lasers.

Figure 5

Temporal evolution of the electron phase-space distribution in the RHC/LHC case. This figure illustrates different snapshots of the electron distribution in phase space at different times labeled on top of each panel. Here, the laser intensity is $a_0=3$. A typical particle is highlighted as a large red dot and its trajectory as the red trail. In the top-right corner of each panel, we plot the spatial profile of the ponderomotive force, which is independent of time in this field configuration.

Figure 6

Temporal evolution of the distribution of electrons in the L/L case. Same as Fig. 5 now for two linearly polarized waves. The ponderomotive force now evolves in time, and the evolution of the phase-space distribution is characterized by two distinct steps: (a)–(c) stretching phase, with a horizontal elongation illustrated by two blue arrows, and (d)–(f) folding phase, with a rotation around an electric field node illustrated by the curved blue arrow.

Figure 7

Temporal evolution of the distribution of electrons in the simplified $\mathit{L}/\mathit{L}$ case. Same as Figs. 5 and 6 but when the ponderomotive force is modeled by a constant piecewise function defined by Eq. (33), in order to clearly separate the two different phases of the electron dynamics.

Figure 8

Analogy between the dynamics of a pendulum and of an electron in two EM waves. Panel (a) shows a sketch of a forced pendulum highlighting the two points of interest: the unstable X point and the stable O point. In panels (b) and (c), we represent the distribution of electrons in phase space after four laser periods ($4T_0$) for two linearly polarized lasers [full relativistic model without approximation, cf. Eq. (10)], below $[a_0=0.1$, panel (b)] and beyond $[a_0=0.8$, panel (c)] the chaos threshold. The equivalents of the O and X points are shown for this configuration.

Figure 9

Electron dynamics in the L/L case in the transition regime from nonchaotic to fully chaotic dynamics. In panel (a), a particle (in red) on a trapped orbit during the folding phase (black dot-dashed line) then follows a ballistic motion from (${\theta }_i, v_{{\theta }_i}$) to (${\theta }_f, v_{{\theta }_i}$) during the stretching phase. If it is fast enough and/or sufficiently close to the separatrix, it can cross an X point along the way and thus move to a neighboring bucket. This is possible for particles within the gray area, called the mixed region. In panels (b) and (c), we represent the distribution of electrons in phase space after $4T_0$ for two linearly polarized lasers [fully relativistic model without approximation; cf. Eq. (10)]: (b) $a_0=0.3$ and (c) $a_0=0.5$. The blue envelope is the separatrix. The black dot-dashed lines show the limit orbit that separates the distribution into two areas: a stable island and a mixed region. It has been obtained with the analytical model presented in the text.

Figure 10

Derivation of the chaos threshold. In panel (a), the minimum laser wave amplitude $a_0^m$ required for a particle to move to a neighboring bucket is plotted in color code, as a function of ${\theta }_i$ and ${\theta }_0$ [Eq. (37)]. ${\theta }_i$ is always smaller than ${\theta }_0$, so that the bottom-right part is forbidden. In addition to the color scale, we plot as plain white lines different isocontours corresponding to $a_0=0.2,0.4,0.6,0.8$. For a given amplitude (e.g., $a_0=0.2$), the parameter ${\theta }_{0,l}$ of the limit orbit between the stable and mixed areas is obtained by minimizing ${\theta }_0$ along the $a_0$ isocontour. The resulting ${\theta }_{0,l}$ is plotted as a function of $a_0$ in panel (b). ${\theta }_{0,l}$ tends to $\pi$ as $a_0$ decreases, and is no longer defined below the chaos threshold ($a_0^T\sim 0.159$).

Figure 11

Particle trajectories as a random walk. In panel (a), four particle trajectories are plotted with different shades of blue. The magnetic nodes are drawn as black dashed lines. For the bottom dark blue trajectory, we highlight its current buckets (dark blue dashed line) as well as the positions and times at which jumps from one bucket to another occur (white dots). At the end of every $T_0/2$ period, a particle can either cross a boundary and then change bucket (events $\pm 1$) or stay in its bucket (event 0). We report in panel (b) the proportions of the three different outcomes summed over a whole simulation for ${10}^5$ particles and $a_0=3$. Finally, panel (c) displays the probability tree discussed in the text, illustrated with the pendulum analogy.

Figure 12

Distribution of particle positions for $n=18$. In panel (a), the electron positions are plotted as a $1\mathrm{D}$ histogram at $t=9T_0$ for the same simulation as in Fig. 11. Each histogram bar corresponds to a magnetic bucket. On top of the histogram is plotted a red line corresponding to the theoretical Gaussian distribution written in Eq. (42). In panel (b), we plot the evolution of the full width at half-maximum (FWHM) of the distribution of panel (a) along time as well as a function varying as $\sqrt{t}$.

Figure 13

Lorentz transformation for the analysis of non-counter-propagating configurations in the laboratory frame. We define $\phi =\alpha /2$, as displayed in this figure. For two counterpropagating waves, $\alpha =\pi$ and $\phi =\pi /2$. The $y$ axis is normal to the figure plane.

Figure 14

Spatial profile of the different forces at various times, for two angles of incidence. For each panel, we plot three forces as a function of position in the boosted frame. The different times and ${\gamma }_b$ are reported in the labels. The vertical dashed lines indicate the positions of magnetic field nodes. In the non-counter-propagating case, they alternatively switch from strong to weak and vice versa every half laser period.

Figure 15

Temporal evolution of the phase-space distribution of particles in the L/L case in oblique incidence. Same as Figs. 5 and 6 but with an angle of incidence $\phi ={30}^{\circ }$. Each panel corresponds to a snapshot of the distribution in phase space for $a_0=3$ at different times. On top of each panel, we plot the corresponding spatial profile of $F_{\text{tot}}$, with different shades of blue to highlight strong (dark) and weak (light) nodes.

Figure 16

Phase-space distribution of particles in L/L configuration at a fixed time for different angles of incidence. Each panel represents the electron distribution in phase space at $t=2.8T_0^{\prime }$ and $a_0=3$ for different angles of incidence indicated in the labels. As before, the different colors encode the initial positions within buckets.

Figure 17

Phase-space distribution of particles in RHC/LHC configuration at a fixed time, for different angles of incidence. Same as Fig. 16 but for a different laser polarization.