Probing the Longitudinal Momentum Spread of the Electron Wave Packet at the Tunnel Exit

We present an ellipticity-resolved study of momentum distributions arising from strong-field ionization of helium. The influence of the ion potential on the departing electron is considered within a semiclassical model consisting of an initial tunneling step and subsequent classical propagation. We find that the momentum distribution can be explained by including the longitudinal momentum spread of the electron at the exit from the tunnel. Our combined experimental and theoretical study provides an estimate of this momentum spread.

Figure 1

Ion momentum distributions and angular distributions in the polarization plane for different ellipticities. The major polarization axis is the $x$ axis, the minor polarization axis is the $y$ axis, and ${\varphi }_p=\arctan (p_y/p_x)$ is the angle of the ion momentum $\mathbf{p}=(p_x,p_y)$. For each panel, the presented data (momentum or angular distribution) are integrated over an interval of 0.05 around the ellipticity value indicated in the headline. The offset angle $\Delta \theta$, indicated in the panel for $\epsilon =0.4$, is defined as the angle between the peak in the angular distributions and the minor polarization axis of the laser pulse.

Figure 2

Ion angular distributions. (a) Ellipticity-resolved angular distributions: the experiment (denoted by “Data”), the simulation with initial longitudinal momentum spread ${\sigma }_{\parallel }=0$, and the simulation with ellipticity-dependent ${\sigma }_{\parallel }$ that yields the smallest square error as compared to the experiment (whose values are given in Fig. 3); see the text. The counts of the images are normalized for each fixed value of ellipticity in order to increase visibility. (b) Angular distributions (line cuts through a) for two different ellipticity values. The simulation with ${\sigma }_{\parallel }=0$ (green curve) does not yield four peaks as in the experiment (red markers). The blue curve shows the simulation that yields the least square error; see the text and Fig. 3.

Figure 3

The blue stars display the value for initial ${\sigma }_{\parallel }$, chosen in semiclassical calculations, that yields the least square error with respect to the experimental angular distributions from Fig. 2b. The red circles are the experimental data for the final momentum spread in the $x$ direction, showing a good agreement with the final longitudinal spread ${\sigma }_{\parallel }^{\mathrm{final}}=\sqrt{3\omega /2{\gamma }^3(1-{\epsilon }^2)}$ as a function of ellipticity predicted by Ref. 31, depicted as a solid black line. For comparison, the values for the final momentum spread in the perpendicular ${\sigma }_{\perp }=\sqrt{\omega /2\gamma }$ (black dashed line) and parallel direction ${\sigma }_{\parallel }^{\mathrm{final}}=\sqrt{3\omega /2{\gamma }^3}$ (black dot-dashed line) introduced in Ref. 19 are also shown.