Generalized eikonal approximation for strong-field ionization

We develop the eikonal perturbation theory to describe the strong-field ionization by finite laser pulses. This approach in the first order with respect to the binding potential (the so-called generalized eikonal approximation) avoids a singularity at the potential center. Thus, in contrast to the ordinary eikonal approximation, it allows one to treat rescattering phenomena in terms of quantum trajectories. We demonstrate how the first Born approximation and its domain of validity follow from eikonal perturbation theory. Using this approach, we study the coherent interference patterns in photoelectron energy spectra and their modifications induced by the interaction of photoelectrons with the atomic potential. Along with these first results, we discuss the prospects of using the generalized eikonal approximation to study strong-field ionization from multicentered atomic systems and to study other strong-field phenomena.

Figure 1

Positions of the saddle points ${\varphi }_s$ as a function of the kinetic energy of electrons $E_{\mathit{p}}$, calculated from Eq. (101). Only these saddle points which satisfy (104) are plotted. The saddle points with the same imaginary part are marked either as solid or dashed lines. The parameters of the driving laser field [described by Eqs. (69) and (86)] are ${\omega }_{\mathrm{L}}=1.55$ eV, $N_{\mathrm{rep}}=3$, and $I=3.125\times {10}^{13}$ W/${\mathrm{cm}}^2$. The final electrons are detected asymptotically at the polar angle ${\theta }_{\mathit{p}}=0.2\pi$.

Figure 2

Shape function $f_{\mathcal{E}}\left(\varphi \right)$ for a triple $\left(N_{\mathrm{rep}}=3\right)$ laser pulse defined by Eq. (86). In the lower panels, the corresponding shape functions $f_A\left(\varphi \right)$ and $f_{\alpha }\left(\varphi \right)$ are displayed [Eqs. (74) and (82), respectively]. We have marked the real parts of saddle points, $\mathrm{Re}{\varphi }_s$, as vertical lines. While the thin black lines correspond to the position of these saddle points which contribute very little to the probability amplitude of ionization (105), the major contribution there comes from the saddle points marked as the thick (both solid blue and dashed red) lines. The positions of $\mathrm{Re}{\varphi }_s$ are for $E_{\mathit{p}}\approx 12U_p$ and ${\theta }_{\mathit{p}}=0.2\pi$.

Figure 3

Dependence of the real (upper panel) and imaginary (lower panel) parts of $G(\mathit{p},{\varphi }_s)$ [Eq. (92)] on the photoionized electron kinetic energy, $E_{\mathit{p}}$, calculated for the same parameters as in Fig. 1. Only these saddle points are accounted for which significantly contribute to the probability amplitude of ionization (105).

Figure 4

Same as in Fig. 3 but for the function $G^{\prime \prime }(\mathit{p},{\varphi }_s)$.

Figure 5

Energy spectra of photoelectrons (113) ionized by the pulse with the ${\sin }^2$ envelope (86). The frequency of the laser field is taken ${\omega }_{\mathrm{L}}=1.55$ eV and its mean intensity is $I=3.125\times {10}^{13}$ W/${\mathrm{cm}}^2$. The envelope of the spectra (solid black line) corresponds to a one-cycle driving pulse $\left(N_{\mathrm{rep}}=1\right)$. Other curves correspond to a sequence of either two one-cycle $(N_{\mathrm{rep}}=2$; dashed red line) or three one-cycle $(N_{\mathrm{rep}}=3$; solid blue line) driving pulses. All results have been divided by $N_{\mathrm{rep}}^2$. They have also been multiplied by $e^{0.6E_{\mathit{p}}/{\omega }_{\mathrm{L}}}$ to magnify the main features of the distributions. The results in the upper frame have been obtained by performing the integral in Eq. (98) exactly. The results in the lower frame have been obtained using the saddle-point method (105). The major features of the mirror-reflected distributions are the same, except that they differ in magnitude.

Figure 6

In the top panel, we show a portion of the energy spectrum presented in the upper frame of Fig. 5. Only the curve for $N_{\mathrm{rep}}=3$ is plotted, and the results are not scaled by $N_{\mathrm{rep}}^2$. Vertical lines mark the energies at which we observe the main maxima. The same but for $N_{\mathrm{rep}}=10$ is plotted in the middle panel. Note that in both cases the main maxima occur at the exact same energies of the final electron. At those energies the function $F\left(\mathit{p}\right)$, drawn in the bottom panel as the solid blue line, takes on integer values. Note that $F\left(\mathit{p}\right)$, in contrast to the dashed black line, is not a linear function of its argument. Therefore, the peaks in the energy distribution of photoelectrons are not equally spaced.

Figure 7

Shape functions for the electric field, $f_{\mathcal{E}}\left(\varphi \right)$, and the electromagnetic potential, $f_A\left(\varphi \right)$, for the ${sin}^2$ envelope, Eq. (114), and for $N_{\mathrm{rep}}=3$ and $N_{\mathrm{osc}}=16$.

Figure 8

Energy spectra of photoelectrons (113) ionized by the train of pulses with the ${\sin }^2$ envelope (114) for $N_{\mathrm{osc}}=16$. The frequency of the laser field is taken ${\omega }_{\mathrm{L}}=1.55$ eV, its peak intensity is $I={10}^{14}$ W/${\mathrm{cm}}^2$, and the photoelectron emission angles are ${\theta }_{\mathit{p}}=0.2\pi$ with an arbitrary ${\phi }_{\mathit{p}}$. The envelope of the spectra (solid black line) corresponds to a 16-cycle driving pulse $\left(N_{\mathrm{rep}}=1\right)$. Other curves correspond to a sequence of either two 16-cycle $(N_{\mathrm{rep}}=2$; dashed red line) or three 16-cycle $(N_{\mathrm{rep}}=3$; solid blue line) driving pulses. All results have been divided by $N_{\mathrm{rep}}^2$. They have also been multiplied by $e^{E_{\mathit{p}}/{\omega }_{\mathrm{L}}}$ to magnify the main features of the distributions. The results have been obtained by performing the integral in Eq. (98) exactly.

Figure 9

Same as in Fig. 5, except that in the lower frame the spectra have been calculated based on the GEA with the application of the quantum trajectory method [Eq. (134)].

Figure 10

Thin black line represents the energy distribution of ionization in the saddle-point Keldysh approximation, Eq. (105), and the thick blue line corresponds to the GEA, Eq. (134). The dashed red line depicts the result for the EA. The upper and the lower panels show the distributions for $N_{\mathrm{rep}}=1$ or 3, respectively. The remaining laser-pulse parameters are the same as in Fig. 5.