Photoelectron combs in ionization: Influence of rescattering and nondipole effects

TL;DR

The paper addresses how photoelectron combs arise in ionization by trains of XUV pulses under nondipole conditions and rescattering. It employs a rigorous TDSE treatment with exact field coupling in a two-dimensional hydrogen model, analyzed through a quasi-relativistic SFA (QRSFA) framework and Fraunhoffer-type interference to interpret interpulse effects. The main findings show comb structures in both momentum and energy distributions, with tilted fringes and angle-dependent peak shifts, spacings near $\omega/N_{\rm osc}$, and partial coherence loss due to rescattering; time delays between pulses further control comb density, and strong-field corrections reveal limitations of dipole and first-order nondipole approximations. These results have implications for attosecond metrology and XUV-pump/XUV-probe experiments where precise, angle-resolved electron dynamics are essential. $N_{ m rep}$ identical pulses and nondipole contributions lead to angle-dependent energy shifts such as $E_{\bm p}(\varphi_{\bm p}) = E_{\bm p} - \sqrt{2 m_e E_{m p}} \frac{\langle U_p\rangle}{m_e c} \sin \varphi_{\bm p}$ with $\langle U_p\rangle = \frac{3}{16} \frac{e^2 A_0^2}{2 m_e}$, illustrating the role of ponderomotive effects in shaping the spectra.

Abstract

Ionization by a sequence of extreme ultraviolet pulses is investigated based on the rigorous numerical solution of the time-dependent Schrödinger equation, when the driving laser field is treated exactly. This goes beyond the typically used first-order nondipole approximation and reveals the effects of radiation pressure to its full extent. Specifically, we observe the comb structures in both the momentum and the energy distributions of photoelectrons. The comb peaks are shifted, however, depending on the emission angle of electrons. While similar effect is observed already in the first-order nondipole approximation, with increasing the laser field strength the discrepancy with our exact results becomes more pronounced. Also, we observe the additional substructure of the comb peaks arising in the angle-integrated energy distributions of photoelectrons. Finally, as our numerical calculations account for the atomic potential in the entire interaction region, we observe the loss of coherence of comb structures with increasing the number of laser pulses, that we attribute to rescattering.

Figures (11)

• Figure 1: The electric field [panel (a)] and the vector potential [panel (b)] plotted at the origin of coordinate system for the model specified by Eq. \ref{['nd3']}, and for the laser field parameters: $\omega=2E_0$, $|eA_0|=5p_0$, $N_{\rm rep}=1$, and $N_{\rm osc}=2$.
• Figure 2: Photoelectron momentum distributions \ref{['nd6']} in the Cartesian coordinates for the laser pulse represented in Fig. \ref{['Laser00']} [panel (a)] and for the train comprising of three such pulses [panel (b)]. The distributions are presented in the logarithmic scale, where the values smaller than $\varepsilon_p= 10^{-8}$ are eliminated.
• Figure 3: Photoelectron momentum distributions \ref{['nd6']} in polar coordinates for the sequence of five XUV pulses ($N_{\rm rep}=5$) shown in Fig. \ref{['Laser00']}. Panel (a) demonstrates the entire energy distribution. On the other hand, panel (b) represents its mid-energy whereas panel (c) shows its low-energy portions. The distributions are presented in the logarithmic scale, where the values smaller than $\varepsilon_p= 10^{-8}$ are eliminated. The red oscillatory curve in panel (b) indicates the analytical prediction of the major peak maxima, Eq. \ref{['fit-red']}, which for the current parameters fits nicely with the numerical results.
• Figure 4: Energy distributions of photoelectrons emitted either in the direction specified by the polar angle $\varphi_{\bm p}=0$ [panel (a)] or $\varphi_{\bm p}=\pm\pi/4$ [panel (b)]. In panel (b), we plot $\tilde{P}_m(E_{\bm p},\varphi_{\bm p}=\pi/4,T_{\rm f})$ in the upper frame and $-\tilde{P}_m(E_{\bm p},\varphi_{\bm p}=-\pi/4,T_{\rm f})$ in the lower frame. Both frames are separated by the solid green line. Moreover, the presented spectra concern ionization driven by either a single XUV pulse (solid blue line) or a train comprising of two (dashed pink line), three (solid red line), and five (solid black line) identical XUV pulses, each of them being represented in Fig. \ref{['Laser00']}. Here, we plot the mid- to high-energy portions of the spectra. All spectra are divided by $N_{\rm rep}^2$.
• Figure 5: Same as in Fig. \ref{['combs1C2a5phi0K']} but for the low-energy portions of the distributions.