Direct Visualization of Laser-Driven Electron Multiple Scattering and Tunneling Distance in Strong-Field Ionization

Using a simple model of strong-field ionization of atoms that generalizes the well-known 3-step model from 1D to 3D, we show that the experimental photoelectron angular distributions resulting from laser ionization of xenon and argon display prominent structures that correspond to electrons that pass by their parent ion more than once before strongly scattering. The shape of these structures can be associated with the specific number of times the electron is driven past its parent ion in the laser field before scattering. Furthermore, a careful analysis of the cutoff energy of the structures allows us to experimentally measure the distance between the electron and ion at the moment of tunnel ionization. This work provides new physical insight into how atoms ionize in strong laser fields and has implications for further efforts to extract atomic and molecular dynamics from strong-field physics.

Figure 1

(a) The photoelectron angular distribution resulting from the ionization of argon gas with a 1300 nm, $7.5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, 40 fs laser pulse. The primary spider structure (minima shown with white lines) results from the interference between directly ionized electrons and those that are driven back to scatter from the Coulomb potential. (b) The newly observed inner structure (minima shown with red lines) results from electron trajectories that scatter from the Coulomb potential on their second reencounter with the ion and interfere with unscattered trajectories. (a)–(e) Simple laser-driven electron dynamics during ionization explain the primary and inner spider structures.

Figure 2

The PSW model. The superposition of a plane wave (a) and a spherical wave (b) generates an interference pattern (c) that has the same shape as the spider structures. The phase of the plane wave is given by ${\Psi }_{\mathrm{plane}}e^{ik{P}_{\parallel }}$, while the phase of the spherical wave is given by ${\Psi }_{\mathrm{spherical}}e^{ik{P}_{\mathrm{total}}}$, where $P_{\parallel }$ is the momentum parallel to the laser polarization, $P_{\mathrm{total}}=\sqrt{P_{\parallel }^2+P_{\perp }^2}$ is the total momentum, and $k$ is the modulation frequency of the plane and spherical waves. Only the real part of the complex wave is shown in panels (a) and (b), while $|\Psi {|}^2$ is shown in panel (c). (d) The minima of the spider structure calculated using the plane-spherical model (white lines) match the experimental data across a broad range of wavelengths and intensities (shown: 1300 nm, $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, 40 fs driving laser).

Figure 3

(a) Depending on when an electron tunnels, it will be driven past the ion a specific number of times. (b) Quasiclassical action obtained by integrating each electron trajectory from when it is born to when it revisits the ion for the second-to-last time versus the final momentum parallel to the laser polarization ($P_{\parallel }$). The frequency of the plane and spherical waves is given by the slope of this graph. The “steps” at low momenta correspond to regions of multiple rescattering. (c) Experimental photoelectron distribution from argon ionized by a $1.3\mu \mathrm{m}$, $7.5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ laser shows a clear boundary between the high momentum region (one revisit) and the low-momentum region where scattering can take place during a subsequent revisit. (d) Photoelectron distribution from xenon using a $2.0\mu \mathrm{m}$, $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ laser displays two boundaries that correspond to the second and fourth revisits. (e) Time-dependent Schrödinger equation (TDSE) calculations reproduce the low-momenta structures in xenon. (f) Single-cycle PSW model, including scattering on the first revisit (outer spiders) as well as the second and fourth revisits (inner spider structures).

Figure 4

(a) In the presence of a strong laser field ($5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$), the atomic Coulomb potential is deformed, enabling tunnel ionization. (b) By examining the classical equations of motion, we plot the theoretical cutoff between trajectories that revisit the ion once and those that rescatter several times (lines). If the tunnel distance is included, the theoretically predicted multiple rescattering cutoffs agree with the experimentally observed cutoffs (points). The experimental intensity was calibrated from the $2U_P$ cutoff. The error in the energy of the multiple rescattering cutoff was assumed to be 0.02 atomic units of momentum.