Numerical Detector Theory for the Longitudinal Momentum Distribution of the Electron in Strong Field Ionization

The lack of analytical solutions for the exit momentum in the laser-driven tunneling theory is a well-recognized problem in strong field physics. Theoretical studies of electron momentum distributions in the neighborhood of the tunneling exit depend heavily on ad hoc assumptions. In this Letter, we apply a new numerical method to study the exiting electron’s longitudinal momentum distribution under intense short-pulse laser excitation. We present the first realizations of the dynamic behavior of an electron near the so-called tunneling exit region without adopting a tunneling approximation.

Figure 1

The final momentum spread in the $x$ direction. In this graph, we show ${\sigma }_x^{\mathrm{ff}}$ values of a theory line (black solid line) [14], experiment data points (red circles) [16], and the SENE result using grid map step size $dL=0.4\mathrm{a}.\mathrm{u}.$ (blue triangles). At $ϵ=0.2$, 0.5, and 0.8, we also show the SENE results using $dL=0.2$ as a “theoretical error bar.” The results of $dL=0.4$ and 0.2 converge at all three ellipticities.

Figure 2

The circle of numerical detectors [21, 22] with radius $R_d$ is shown, as well as outgoing classical particle trajectories that were initiated with the momentum values determined by the detectors. The trajectories continue to be fully affected by both laser action and ionic Coulomb attraction as the particles propagate outward to actual detection.

Figure 3

Rows from top to bottom: $ϵ=0.2$, 0.5. Columns from left to right: cumulative initial momentum distributions in the polarization (x-y) plane at NDs through whole laser pulse of (1) helium, (2) krypton. The peak laser intensities used are for helium $0.8\mathrm{PW}/{\mathrm{cm}}^2$, and for krypton $0.12\mathrm{PW}/{\mathrm{cm}}^2$.

Figure 4

Time-resolved values of ${\overline{p}}_{xt}$, ${\overline{p}}_{yt}$, and $T$ is one laser period. The discrete step size is $T/8$. The width of the momentum distribution in each time window is plotted as vertical error bars. Columns from left to right: (i) helium and (ii) krypton. All single plots show the time-dependent parameters’ changes over the center two and a half cycles of the laser pulse. We also plot unit field oscillation in both the $x$ and $y$ directions.

Figure 5

This figure shows how the cumulative $\sigma (p_x)$ and $\sigma (p_y)$ change vs ellipticities for helium and krypton atoms. In a near-circularly polarized laser beam, two standard deviations converge. We also compare our data of standard deviations in longitudinal momentum of helium atoms to the prediction made by the TIPIS model [16].