Electron trajectories associated with laser-driven coherent synchrotron emission at the front surface of overdense plasmas

We present an in-depth analysis of an ultrafast electron trajectory type that produces attosecond electromagnetic pulses in both the reflected and forward directions during normal incidence, relativistic laser-plasma interactions. Our particle-in-cell simulation results show that for a target which is opaque to the frequency of the driving laser pulse the emission trajectory is synchrotronlike but differs significantly from the previously identified figure-eight type which produces bright attosecond bursts exclusively in the reflected direction. The origin and characteristics of this trajectory type are explained in terms of the driving electromagnetic fields, the opacity of the plasma, and the conservation of canonical momentum.

Figure 1

Initial (a) laser and (b) plasma parameters used in the principal PIC simulation of this article.

Figure 2

Transverse electromagnetic fields (a) $E_y(x,t)$ and (b) $B_z(x,t)$ (blue to red) overlaid on the electron density $n_e(x,t)$ (gray to black) during the simulated interaction. Since the target is overdense, the transverse fields in front of its surface are a superposition of incoming and reflected waves and the long-wavelength fields tend towards zero inside the plasma. (c) Longitudinal $E_x(x,t)$ field, which is caused by deviations from local charge neutrality arising from the longitudinal motion of the particles.

Figure 3

Electron density overlaid with the (a) longitudinal $j_x(x,t)$ and (b) transverse $j_y(x,t)$ current density components. The overall motion of the front surface consists of transverse oscillations once every laser period and longitudinal oscillations twice every laser period. Six individual macroelectron trajectories are selected from different regions across the width of two successive electron oscillations. Also shown is their motion in the (c) $xt$ and (d) $xy$ planes.

Figure 4

Collective electron dynamics and attosecond emission surrounding the (a) reflected and (b) forward emission points. The electron density $n_e(x,t)$ (gray to black) is shown at the front surface of the plasma during the peak of the simulated interaction. Overlaid (magenta) is the square of the transverse electric field $|e{E}_{y,\mathrm{filt}}/m_{e}c{\omega }_L{|}^2$, where $E_{y,\mathrm{filt}}$ is the transverse electric field filtered in the range $\omega /{\omega }_L=100–200$.

Figure 5

(a) Trajectory (shown as dots) of the representative macroelectron at equally spaced time intervals from $t/T_L=8$ to $t/T_L=10.5$. Also shown are the (b) velocity $v_{x,y}\left(t\right)$ and (c) acceleration $a_{x,y}\left(t\right)$ components of the highlighted particle. At $t/T_L\approx 9.64$ and 10, the electron bunch has both a relativistic longitudinal velocity and a large transverse acceleration, which results in the emission of synchrotron radiation.

Figure 6

Comparison between the attosecond pulses obtained directly from the PIC simulation (orange dotted line) and those calculated from the single-particle trajectory (blue solid line): (a) reflected pulse at position $x/{\lambda }_L=-8$ and (b) forward pulse collected at $x/{\lambda }_L=9.1$. In both cases, the pulses are filtered to only include frequencies in the range $\omega /{\omega }_L=100–200$.

Figure 7

(a)–(c) Particle trajectory in the $xy$ plane overlaid with arrows that represent the magnitude and direction of the six acceleration terms defined by Eq. (5) at each point. To help interpret these results, (d) $E_x(x,t)$, (e) $E_y(x,t)$, and (f) $B_z(x,t)$ (blue to red) are shown in the vicinity of the emitting particle over the same time period (yellow line).

Figure 8

PIC code simulation results showing emitting electron dynamics in a relativistically transparent target. The electron density (gray to black) is overlaid with the (a) $E_x(x,t)$, (b) $E_y(x,t)$, and (c) $B_z(x,t)$ field components (blue to red). The position of one macroelectron which typifies the collective motion is plotted with the yellow line. The black dots in (d) show the trajectory of this selected particle in the $xy$ plane and the yellow arrows represent the magnitude and direction of the acceleration vector at each point. In this case, the particle exhibits a figure-eight motion. Also shown are the (e) velocity and (f) acceleration components of this particle.