Theory of low-energy photoelectrons in strong-field ionization by laser pulses with large ellipticity

We theoretically analyze the low-energy electrons appearing in strong-field ionization of atomic and molecular targets by laser pulses with large ellipticity. We present the semiclassical model used to calculate the photoelectron momentum distributions and the derivations leading to a parameter introduced in a recent combined theoretical and experimental work on naphthalene [D. Dimitrovski et al., Phys. Rev. Lett. 113, 103005 (2014)]. The parameter quantifies both the possibility to observe such low-energy electrons and their relative impact in the photoelectron momentum distributions. The numerical value of this parameter is calculated for various cases of strong-field ionization by elliptically polarized pulses available in the literature; in addition, the possibility to detect low-energy electrons for large $\epsilon$ and different targets, including simple atoms, is discussed.

Figure 1

Calculated photoelectron energy distributions for ionization of naphthalene molecules ($I_p=0.295$ a.u.) for the parameters of Ref. [22]: $F_0=0.045$ a.u., $\epsilon =1/\sqrt{2}$, and wavelength $\lambda =800$ nm. (a) Total energy distributions. (b) Energy distributions for a thin cut of momenta involving the $yz$ plane of polarization and transverse momenta with $|p_x|<0.015$ a.u. The blue curve with circles shows the calculation with the full potential $V\left(\mathbf{r}\right)$ of Eq. (4), while the black curve with squares shows the calculation only involving the Coulomb potential $V_{\text{C}}\left(\mathbf{r}\right)$. The arrow points to the energy obtained in the simple man's model (1).

Figure 2

Fixed-in-space naphthalene: Tunnel exit points that lead to production of low-energy electrons (dots). The orientation of the molecular axes is shown along with the coordinates in the molecular center of mass. The molecule and the exit points are shown on the same scales. The field ellipse is sketched by the full (red) curve.

Figure 3

Theoretical analysis of the low-energy spectra. The origin of the low-energy (smaller than 0.05 a.u.) electrons (black open circles) and Rydberg electrons (red open squares): (a) in the plane spanned by the component of the initial momentum of the electrons perpendicular to the polarization plane of the laser pulse ($p_{0x}$), and the time during the evolution of the laser pulse (in cycles), where $T_p=2\pi /\omega$ is the duration of one cycle of the field; (b) magnification of (a) for the central half-cycle of the component of the electric field along the most polarizable axis of the molecule; (c) in the plane spanned by the components of the initial momentum along the major axis of the pulse ($p_{0y}$) and the component of the initial momentum along the minor axis of the pulse ($p_{0z}$) for the central half-cycle of the pulse. See text for details.

Figure 4

Total energy distributions of electrons as a function of laser pulse parameters ($\omega$ and $\epsilon$) and parameters of the target ($I_p$ and $\alpha$). In all panels the blue curves with circles denote the total energy distribution for the experimental parameters of Ref. [22]: $F_0=0.045$ a.u. (constant in all panels), $\epsilon =1/\sqrt{2}$, $I_p=0.295$ a.u., and wavelength $\lambda =800$ nm. In each panel, only one parameter is changed, keeping all other parameters fixed: (a) the dependence on $\lambda$; (b) the dependence on $\epsilon$; (c) the dependence on $I_p$; and (d) the dependence on polarizability. The black curve with squares in (d) is obtained by halving the matrix elements of the polarizability tensor for naphthalene and its cation. The arrows point to the energy obtained in the simple man's model (1) for different parameters.