Coulomb-induced asymmetry on photoelectron momentum distributions in orthogonal two-color laser fields

By numerically solving the time-dependent Schrödinger equation, we study the ionization of helium atoms with different forms of the atomic potential, including the soft-core potential, the model potential, and the short-range potential, in orthogonally polarized two-color laser fields. We show that in the case of the long-range atomic potentials (the soft-core potential and the model potential), the photoelectron momentum distributions exhibit asymmetric structures, while for the short-range potential, they present nearly symmetric structures. Besides, the asymmetric structures are sensitive to the specific forms of the atomic potential. We find that the asymmetric structures are more pronounced for the lower laser intensity and/or the shorter laser wavelength. Our results may provide a reference for further analytical studies of strong-field ionization of complex atoms and molecules, where the Coulomb effects are included for quantitative descriptions.

Figure 1

(a) Three typical forms of the atomic potential [see Eqs. (1a)–(1c)]: the short-range potential $V_{\mathrm{short}}$ (dashed black), the model potential $V_{\mathrm{model}}$ (solid red), and the soft-core potential $V_{\mathrm{soft}}$ (dash-dotted blue). (b) Schematic diagram of the electric fields of the OTC laser fields [see Eq. (2)]. Here, the peak intensity is $5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and the carrier frequency is $0.057\mathrm{a}.\mathrm{u}.$. A trapezoidal pulse envelope is used with a total duration of 10 optical cycles of the fundamental field, switched on and off linearly over two cycles. The total electric field is plotted in green, while the fundamental field and its second-order harmonic field are shown as red and blue lines, respectively. (c1) Amplitude of the fundamental (red) and the second harmonic (blue) fields, and (c2) the corresponding drift momenta caused by the laser fields, $p_x\left(t\right)=-A_x\left(t\right)$ and $p_y\left(t\right)=-A_y\left(t\right)$, within one optical cycle of the fundamental field. Four subcycles, I, II, III, and IV are marked in (c1).

Figure 2

Photoelectron momentum distributions in the $xy$ polarization plane with $p_z=0$ at different laser intensities for the soft-core potential (top row), the model potential (middle row), and the short-range potential (bottom row). The laser wavelength is 800 nm. The black curves in (a1) and (a3) denote the field-driven photoelectron momenta $\vec{p}=-\vec{A}\left(t\right)$ within one optical cycle of the fundamental field. Panels in the column from left to right corresponds to the intensities of $3\times {10}^{14}$, $5\times {10}^{14}$, and $6\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, respectively. The tilt angle $\theta$ is the angle between the $p_x$ axis and the most probable direction of distribution, as marked in panel (a2).

Figure 3

Photoelectron momentum distributions. The conditions used are the same as the case shown in the middle column in Fig. 2 except that the wavelengths in the left and right columns are 600 and 1000 nm, respectively.

Figure 4

Up-down symmetry parameter, $\eta$ as a function of the laser intensity for the soft-core potential (dashed blue lines) and the model potential (solid red lines) with three laser wavelengths: 600 nm (square markers), 800 nm (circle markers), and 1000 nm (triangle markers).

Figure 5

Tilt angle $\theta$ as a function of the laser ellipticity $\varepsilon$ for three atomic potentials: the soft-core potential (blue circle), the model potential (red square), and the short-range potential (black triangle). For comparison, the pink hexagon line is the prediction given by Eq. (10). The laser intensity is $5\times {10}^{14}$ $\mathrm{W}/{\mathrm{cm}}^2$ and the laser wavelength is 800 nm.