Asymmetries in strong-field photoionization by few-cycle laser pulses: Kinetic-energy spectra and semiclassical explanation of the asymmetries of fast and slow electrons

Using numerical solutions of a time-dependent Schrödinger equation for a hydrogen atom in a linearly polarized few-cycle laser field, we calculate the left-right photoelectron kinetic-energy spectra measured by two opposing detectors placed along the laser polarization vector, with laser focus in the center. The fastest electrons show huge asymmetries strongly dependent on the laser carrier-envelope (CE) phase which confirms the recent theoretical results [D. B. Milosevic et al., Opt. Express 11, 1418 (2003)], obtained from a modified strong field approximation model which includes rescattering by the Coulomb potential. This asymmetry can also be explained by a simple semiclassical model in which the electron after tunneling through a potential barrier returns to the proton and is elastically backscattered in the presence of the laser field thus acquiring energy close $10U_p$ where $U_p$ is the electron ponderomotive energy in the laser field. We also present a semiclassical interpretation of counterintuitive left-right asymmetries of slow electrons discussed in our previous work [Phys. Rev. A. 70, 013815 (2004)]. Our analysis shows that the Coulomb attraction from the proton must be included in the standard tunneling model in order to account for the CE phase dependent angular asymmetry seen in our previous numerical calculations.

Figure 1

Electric field $E\left(t\right)∕E_0$ and its envelope ${\epsilon }_0\left(t\right)∕E_0$ as function of time in cycles for CE phases: (a) $\varphi =0$; (b) $\varphi =\pi ∕2$; (c) $\varphi =\pi$. The central wavelength is $\lambda =800\mathrm{nm}$ and the intensity profile half-width is ${\tau }_p=3.9\mathrm{fs}$. The bullet in (a) designates the tunneling time leading to the trajectory which ends with the electron energy $10U_p$.

Figure 2

Left and right ATI electron spectra $S_{\text{left}}\left(E_e\right)$ (dotted line) and $S_{\text{right}}\left(E_e\right)$ (solid line) defined by Eq. (6) for laser pulses having duration (FWHM) ${\tau }_p=3.9\mathrm{fs}$ for the laser CE phase $\varphi =0$ and for laser intensities (a) $6\times {10}^{13}$; (b) ${10}^{14}$; and (c) $2\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 3

Same as in Fig. 2 but for the laser CE phase $\varphi =\pi ∕2$.

Figure 4

Low energy part of the ATI spectra shown in Figs. 2a, 3a.

Figure 5

Left and right ATI electron spectra $S_{\text{left}}\left(E_e\right)$ (dotted line) and $S_{\text{right}}\left(E_e\right)$ (solid line) defined by Eq. (6) for $I={10}^{14}$, $\varphi =0$ and for various pulse durations ranging from ${\tau }_p=5\text{to}{\tau }_p=6.0\mathrm{fs}$.

Figure 6

Left and right ATI electron spectra (only its high energy part) $S_{\text{left}}\left(E_e\right)$ (dotted line) and $S_{\text{right}}\left(E_e\right)$ (solid line) defined by Eq. (6) for $I=6\times {10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$, ${\tau }_p=3.6\mathrm{fs}$ and for laser CE phases $\varphi$ ranging from 0° to 67.5°.

Figure 7

For laser CE phases $\varphi$ ranging from 90° to 157.5°. Note that because of symmetry the case of $\varphi =180°$ can be deduced from the $\varphi =0$ case shown in Fig. 6a by interchanging $S_{\text{right}}$ and $S_{\text{left}}$.

Figure 8

Asymmetry coefficient defined by Eq. (12) as function of the CE phase $\varphi$ for the laser intensity $I={10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, ${\tau }_p=3.9\mathrm{fs}$ (dash-dotted line) classical calculation using the quasistatic tunneling rate Eq. (7) (Solid line): classical calculation using the nonadiabatic rate; dotted line: quantum calculation based on the TDSE.

Figure 9

Asymmetry coefficient obtained using the semiclassical model and the nonadiabatic rate for the laser intensity $I={10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, ${\tau }_p=3.9\mathrm{fs}$. Solid line: with Coulomb attraction, using Eq. (8); dashed line: without the Coulomb attraction using Eq. (9).

Figure 10

Asymmetry coefficient obtained using the semiclassical model and the nonadiabatic rate for ${\tau }_p=3.9\mathrm{fs}$ and for the laser intensities: $I=6\times {10}^{13}$, $I={10}^{14}$, and for $I=1.35\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$.