Electron emission perpendicular to the polarization direction in laser-assisted XUV atomic ionization

We present a theoretical study of ionization of the hydrogen atom due to an XUV pulse in the presence of an infrared (IR) laser with both fields linearly polarized in the same direction. In particular, we study the energy distribution of photoelectrons emitted perpendicularly to the polarization direction. As we previously showed in Gramajo et al. [Phys. Rev. A 94, 053404 (2016)] for parallel emission, by means of a very simple semiclassical model which considers electron trajectories born at different ionization times, the electron energy spectrum can be interpreted as the interplay of intra- and intercycle interferences. However, contrary to the case of parallel emission the intracycle interference pattern stems from the coherent superposition of four electron trajectories giving rise to (i) interference of electron trajectories born during the same half cycle (intra-half-cycle interference) and (ii) interference between electron trajectories born during the first half cycle with those born during the second half cycle (inter-half-cycle interference). The intercycle interference is responsible for the formation of the sidebands. We also show that the destructive inter-half-cycle interference for the absorption and emission of an even number of IR laser photons is responsible for the characteristic sidebands in the perpendicular direction separated by twice the IR photon energy. This contrasts with the emission along the polarization axis (all sideband orders are present) since intra-half-cycle interferences do not exist in that case. The intracycle interference pattern works as a modulation of the sidebands and, in the same way, it is modulated by the intra-half-cycle interference pattern. We analyze the dependence of the energy spectrum on the laser intensity and the time delay between the XUV pulse and the IR laser. Finally, we show that our semiclassical simulations are in very good agreement with quantum calculations within the strong-field approximation and the numerical solution of the time-dependent Schrödinger equation, giving rise to nonzero emission, in contraposition to other theories.

Figure 1

Emission times (solutions of Eq. (9)) as the intersection of the two curves $A^2\left(t\right)=A_{L0}^2{sin}^2\left({\omega }_{L}t\right)$ in red solid line and $v_0^2-k_{\rho }^2$ in black solid line for one IR optical cycle. In this particular case, the XUV pulse starts when the potential vector vanishes.

Figure 2

Buildup of the interference pattern following the SCM for $N=2$. (a) Intra-half-cycle interference pattern given by $G\left(k_{\rho }\right)$ in thick red line, intercycle pattern given by the factor $B\left(k_{\rho }\right)$ in thin dotted green (light grey) line and inter-half-cycle pattern given by the factor $H\left(k_{\rho }\right)$ in thin blue (grey) line. (b) Intracycle pattern given by the factor $F\left(k_{\rho }\right)$ in thick gray line and total interference pattern $F\left(k_{\rho }\right)B\left(k_{\rho }\right)=G\left(k_{\rho }\right)H\left(k_{\rho }\right)$ in thin black line. Vertical lines depict the positions of the sidebands $E_{\ell }$ of Eq. (19). The IR laser parameters are $F_{L0}=0.05$, and ${\omega }_L=0.05$, and the XUV frequency is ${\omega }_X=1.5$.

Figure 3

Photoelectron spectra in the perpendicular direction calculated within (a) the SFA and (b) the TDSE, for different XUV pulse durations ${\tau }_X=T_L/2$ (thick light gray), $T_L$ (thick gray), and $2T_L$ (thin black) and respective time delays ${\mathrm{\Delta }}_X=T_L/4, T_L/2$, and $T_L$. The XUV and IR parameters are the same as in Fig. 2 and ${\tau }_L=5T_L$ and $F_{X0}=0.05$. Vertical lines depict the positions of the sidebands according to Eq. (19). The photoelectron energy distributions have been rescaled for better visualization.

Figure 4

Photoelectron spectra in the perpendicular direction (in arbitrary units) calculated at different laser field strengths within the SCM [(a), (d), and (g)], the SFA [(b), (e), and (h)], and the TDSE [(c), (f), and (i)]. The XUV pulse durations are ${\tau }_X=T_L/2$ [(a)–(c)], ${\tau }_X=T_L$ [(d)–(f)], and ${\tau }_X=2T_L$ [(g)–(i)]. The other XUV and IR parameters as in previous figures. In green dotted line we show the classical boundaries and in black dotted line the $E_{\ell }$ values given by Eq. (19). The photoelectron energy distributions have been rescaled for better visualization.

Figure 5

Photoelectron spectra in the perpendicular direction (in arbitrary units) calculated as a function of the time delay ${\mathrm{\Delta }}_X$ within the SCM [(a), (d), and (g)], the SFA [(b), (e), and (h)], and the TDSE [(c), (f), and (i)]. The XUV pulse durations are ${\tau }_X=T_L/2$ [(a)–(c)], ${\tau }_X=T_L$ [(d)–(f)], and ${\tau }_X=2T_L$ [(g)–(i)]. The other XUV and IR parameters as in previous figures. In dotted line we show the energy values of Eq. (22). The photoelectron energy distributions have been rescaled for better visualization.