Interference Oscillations in the Angular Distribution of Laser-Ionized Electrons near Ionization Threshold

We analyze the two-dimensional momentum distribution of electrons ionized by few-cycle laser pulses in the transition regime from multiphoton absorption to tunneling by solving the time-dependent Schrödinger equation and by a classical-trajectory Monte-Carlo simulation with tunneling (CTMC-T). We find a complex two-dimensional interference pattern that resembles above threshold ionization (ATI) rings at higher energies and displays Ramsauer-Townsend-type diffraction oscillations in the angular distribution near threshold. CTMC-T calculations provide a semiclassical explanation for the dominance of selected partial waves. While the present calculation pertains to hydrogen, we find surprising qualitative agreement with recent experimental data for rare gases [A. Rudenko et al., J. Phys. B 37, L407 (2004)].

Figure 1

Doubly differential electron momentum distributions in cylindrical coordinates $(k_z,k_{\rho })$. The parameters of the field are $\omega =0.05$, $\tau =1005$. In (a) $\gamma =1.34$ ($F_0=0.0377$), (b) $\gamma =0.95$ ($F_0=0.053$), and (c) $\gamma =0.67$ ($F_0=0.075$). In (c) the border between the first and second ATI rings is drawn as a dashed line.

Figure 2

Angular distribution calculated by solving the TDSE (thick solid lines) for $\omega =0.05$, $\tau =1005$, $\gamma =0.67$ ($F_0=0.075$), and $k=0.19$. Dashed lines: square of the Legendre polynomial $P_{l_0}(\cos {\theta }_k),$ $l_0=8$.

Figure 3

Partial ionization probability, $p_l$, as a function of the angular momentum $l$ for different spectral regions indicated in the figure. The parameters of the field are $\omega =0.05$, $\tau =1005$, and $\gamma =0.67$ ($F_0=0.075$).

Figure 4

(a) $L$ distribution of classical trajectories for the same parameters as in Fig. 3. (b) Classical angular distribution for $l=8$ ($L\in [8,9]$) in the region $k<0.26$. (c), (d) Classical trajectories of laser-driven electrons (solid lines) and unperturbed Kepler hyperbola of same asymptotic $E$ and $L$ (dashed lines) differing in the number of quiver oscillations on the outgoing path. Interferences occur for emission at different times $t_i$ close to different field maxima.