Semiclassical two-step model for strong-field ionization

We present a semiclassical two-step model for strong-field ionization that accounts for path interferences of tunnel-ionized electrons in the ionic potential beyond perturbation theory. Within the framework of a classical trajectory Monte Carlo representation of the phase-space dynamics, the model employs the semiclassical approximation to the phase of the full quantum propagator in the exit channel. By comparison with the exact numerical solution of the time-dependent Schrödinger equation for strong-field ionization of hydrogen, we show that for suitable choices of the momentum distribution after the first tunneling step, the model yields good quantitative agreement with the full quantum simulation. The two-dimensional photoelectron momentum distributions, the energy spectra, and the angular distributions are found to be in good agreement with the corresponding quantum results. Specifically, the model quantitatively reproduces the fanlike interference patterns in the low-energy part of the two-dimensional momentum distributions, as well as the modulations in the photoelectron angular distributions.

Figure 1

Vectorial momentum distributions for the H atom ionized by a laser pulse with a duration of $n=8$ cycles, wavelength of $\lambda =800\mathrm{nm}$, and peak intensity of $I=0.9\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ obtained from (a) the QTMC model, (b) solution of the TDSE, and (c) the present SCTS model. The distributions are normalized to the total ionization yield. A logarithmic color scale in arbitrary units is used. The laser field is linearly polarized along the $z$ axis.

Figure 2

Magnification of Fig. 1 for $|k_z|,|k_{\perp }|<0.3$ a.u.

Figure 3

Vectorial momentum distribution from the present SCTS model for low-energy electrons without (a),(c) and with (b),(d) inclusion of the postpulse Coulomb phase ${\tilde{\mathrm{\Phi }}}_f^C\left({\tau }_f\right)$ [Eq. (40)] for the eight-cycle pulse of Fig. 1 (a),(b) and a single-cycle pulse (c),(d) with all other laser parameters identical. The distributions are normalized to the total ionization yield. A logarithmic color scale in arbitrary units is used.

Figure 4

Energy spectra (a),(c),(e) and angular distributions (b),(d),(f) of the photoelectrons for ionization of H at an intensity of $9\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ and a pulse duration of eight cycles obtained from the QTMC model [thin (magenta) curve with solid circles], the SCTS model [solid (blue) curve], and TDSE [thick (green) curve]. The distributions [(a),(b)], [(c),(d)], and [(e),(f)] correspond to the wavelengths of 800, 1200, and 1600 nm, with Keldysh parameters of 1.12, 0.75, and 0.56, respectively. The energy spectra are normalized to the peak value; the angular distributions are normalized to the total ionization yield and show the spectrum for electrons with asymptotic energies $E<U_p$. The energy equal to $U_p$ is shown by arrows in panels (a) and (c).

Figure 5

Angular distributions for low-energy electrons (innermost fanlike structure; cf. Fig. 2): (a) $E<0.022$ a.u. for $\lambda =800\mathrm{nm}$, (b) $E<0.031$ a.u. for $\lambda =1200\mathrm{nm}$, and (c) $E<0.036$ a.u. for $\lambda =1600$ nm. Cutoff energies have been determined from TDSE results.

Figure 6

Vectorial momentum distributions obtained from (a) SCTS model with zero initial parallel velocity, (b) solution of the TDSE, and (c) the SCTS model with nonzero initial parallel velocity. The parameters are the same as in Fig. 1. Panel (a) is the same as Fig. 1. A logarithmic color scale in arbitrary units is used.

Figure 7

Same as Fig. 4 for $\lambda =800$ nm with nonzero initial parallel velocity.