Forward- and backward-scattering quantum orbits in above-threshold ionization

High-order above-threshold ionization (ATI) is characterized by the rescattering of the ionized electron off the parent ion. The ATI amplitude can be calculated applying the saddle-point (SP) method and the uniform approximation to a five-dimensional integral over the ionization time, intermediate electron momentum, and rescattering time. The so-obtained amplitude is a coherent sum of amplitudes associated to the two classes of the SP solutions. The low-energy structures in the ATI spectra are determined by the forward-scattering SP solutions, while the high-energy spectra (plateau and cutoff) are determined by the backward-scattering SP solutions. In order to properly describe the intermediate-electron-energy spectra, additional SP solutions should be taken into account. For these solutions, the imaginary part of the rescattering time can be large and, for their explanation, we introduce complex-time quantum orbits.

Figure 1

Examples of the notation $(\alpha ,\beta ,m)$ used to label the backscattering solutions of the saddle-point equations. (a) The solid, dotted, long-dashed, and dot-dashed curves in the right-hand part of the figure ($0\le \mathrm{Re}t\le T$) specify the real part of the backscattering times for the four pairs of orbits with the shortest travel times. In the left-hand part of the figure, the counterpart of each curve identifies the corresponding real part of the ionization times $t_0$. The emitted electron energy in multiples of $U_p$ is plotted on the ordinate, and horizontal lines (at constant energy) relate the real parts of the ionization and backscattering times for the respective orbits. There are infinitely many further solutions that have the real part of the ionization time beyond the left-hand margin of the figure. Their maximal energies $E_{\mathbf{p}}$ converge to $8U_p$. (b) The saddle-point solutions $(\alpha ,\beta ,m)=(\pm 1,1,0)$ which are introduced in the present paper. The curves which correspond to the real part of the ionization time $t_0$ and the rescattering time $t=t_r$ are denoted by arrows and corresponding value of $\alpha$. The curves have been calculated for Ar, for emission in the direction $\theta =0^{\circ }$, and for a linearly polarized laser field having the intensity $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and the wavelength 2000 nm.

Figure 2

Explanation of the classification of the forward-scattering saddle-point solutions. The leftmost red curve, denoted as the ionization time $t_0$, which is the same for all saddle-point solutions, represents the electron kinetic energy $E_{\mathbf{p}}$ in units of $U_p$ (on a logarithmic scale) as a function of $\mathrm{Re}{t}_0/T$. All other curves specify the corresponding real parts of the forward-scattering times, $\mathrm{Re}{t}_r/T$, for five pairs of orbits with the shortest travel times. The orbits are characterized by the multi-index $\nu \mu$, where the index $\mu =0,1,2,...$ denotes a pair of solutions, while the members of each pair are distinguished by the index $\nu =\pm 1$. There are further solutions that have forward-scattering times beyond the right-hand margin of the figure. Their maximal (cutoff) energies $E_{\mathbf{p}}$ decrease. The results are obtained for Ar and for the same laser parameters as in Fig. 1.

Figure 3

The differential ionization rates of Ar as functions of the electron energy in units of $U_p$ for the same laser parameters as in Fig. 1. A logarithmic scale is used and three characteristic regions of the spectrum are denoted: low-energy structures (LES), plateau, and the cutoff. The results are obtained using the uniform approximation with 20 saddle-point solutions $\beta m$ (black solid “bs” curve) and using the saddle-point method with 15 forward-scattering solutions (red dashed “fs” curve).

Figure 4

The differential detachment rates of ${\mathrm{H}}^{-}$ as functions of the electron energy in units of $U_p$. The electron emission angle is $\theta =0^{\circ }$ and the wavelength and intensity of linearly polarized laser field are $10600$ nm and ${10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, respectively. Comparison of the result obtained using the exact method (“ex”, black solid line) [40] with the result obtained using the quantum-orbit formalism with backscattered orbits [39] (“bs”, red dot-dashed line; additional orbits presented in this paper are not taken into account) and with the result obtained using only the forward-scattered orbit $\nu \mu =10$ [35] (“fs”, blue dotted line). The coherent sum of the “fs” and “bs” results is also shown (green dashed line).

Figure 5

Same as in Fig. 4, but for comparison of the results obtained using various approximations: (i) the uniform approximation (UA) with 18 saddle-point solutions $\beta m$ (black solid curve denoted by “all”; $\beta m\ne 10$), (ii) UA with only the solutions having $\beta m=-10$ (blue long-dashed curve), (iii) UA with 17 solutions $\beta m$ (green dot-dashed curve denoted by “rest,” since the solution $\beta m=-10$ is excluded from “all” solutions), and (iv) results obtained using the saddle-point method and only the solution $(\alpha ,\beta ,m)=(1,-1,0)$ [$(-1,-1,0)$] [red dotted (magenta short-dashed) curve].

Figure 6

Backward-scattered quantum-orbit solutions and trajectories for above-threshold detachment of ${\mathrm{H}}^{-}$ for the same laser parameters as in Fig. 4. Left: The $\beta m=-10$ saddle-point solutions for the complex ionization time $t_0$ and backscattering time $t_r$. The $\alpha =+1$ ($\alpha =-1$) solutions are denoted by red dot-dashed (black solid) lines. The electron energy changes from a minimum value to the cutoff value as a continuous parameter along each curve. Right: Real part of the quantum orbits obtained using the saddle-point solutions of the left-hand panel. The electron energy is $E_{\mathbf{p}}=18\omega =2U_p$. The electron is born at $\mathrm{Re}{x}_{18}\left(\mathrm{Re}{t}_0\right)=17.7$ a.u. (15 a.u.) for the $\alpha =+1$ ($\alpha =-1$) solution, then moves further away from the origin and, when the field changes its sign, the electron turns around, moves back to the atom, and backward scatters off it. Finally, it reaches the detector which is in the $\theta =0^{\circ }$ direction. The “exit” of the tunnel and the backscattering point are denoted by the corresponding symbols.

Figure 7

Forward-scattered quantum-orbit solution $\nu \mu =10$ and trajectories for fixed electron energies for above-threshold detachment of ${\mathrm{H}}^{-}$ for the same laser parameters as in Fig. 4. Left: Saddle-point solutions for the complex ionization time $t_0$ (black solid line) and forward-scattering time $t_r$ (red dot-dashed line). The electron energy changes from a minimum value to the cutoff value as a continuum parameter along each curve. Right: Real part of the quantum orbits obtained using the saddle-point solutions of the left-hand panel. The electron energy in units of $\omega$ ($U_p$) is $q=6$ (0.67; black solid line), 9 (1.00; red dotted line), 14 (1.56; green long-dashed line), 18 (2.00; blue dashed line), and 22 (2.45; cyan dot-dashed line). The positions where the electron “exits” from the tunnel and where it forward scatters on the H atom are denoted by the corresponding symbols.

Figure 8

Same as in Fig. 7, but for $\nu \mu =\pm 10$. The trajectories in the right-hand panel are for the electron energy $E_{\mathbf{p}}=1.5U_p$. The rescattering times which correspond to this energy are denoted by diamonds in the left-hand panel. The red dashed trajectory is shifted up by 10 a.u. in order to avoid a visual overlap with the black solid curve.

Figure 9

Same as in Fig. 6, but for the solutions $(\alpha ,\beta ,m)=(+1,1,0)$ (upper panels) and $(-1,1,0)$ (lower panels). The complex time $t_r$ of the solution $(-1,1,0)$ (red dashed line) is presented in the upper left panel, while the corresponding time $t_0$ is shown in the lower left panel.

Figure 10

Same as in Fig. 4, but with added partial contributions of the short-time saddle-point solutions $(\alpha ,\beta ,m)=(\pm 1,1,0)$ and $\nu \mu =\pm 10$. The coherent sum of all saddle-point solutions (circles denoted by “all”; the solution $-10$ is excluded) agree well with the exact solution obtained by numerical integration (solid line with diamonds denoted by “ex”).

Figure 11

Upper panels: Complex-time curves defined by Eq. (24) for the quantum orbits $(\alpha ,\beta ,m)=(\pm 1,1,0)$ and for the electron energy $E_{\mathbf{p}}=1.5U_p$ (compare Fig. 9). Lower panel: Real part of the quantum orbits for the complex time which follows the curves (24) shown in the upper panels. The time $t_C$ on the abscissa is defined as the real or imaginary part of the complex time which changes along the curves $C_j$ (see the text for more explanation).

Figure 12

Same as in the right-hand panels of Figs. 6 and 8, but for the different definition of the quantum orbits and for the electron energy $E_{\mathbf{p}}=1.5U_p$.