High-order above-threshold ionization beyond the first-order Born approximation

In the improved strong-field approximation, which describes high-order above-threshold ionization (HATI), the rescattering of the ionized electron on the parent ion is described within the first-order Born approximation. The low-frequency approximation for laser-assisted scattering goes beyond the first Born approximation. In the present paper, we derive the low-frequency approximation for HATI. The rescattering amplitude in the first Born approximation is replaced by the exact scattering amplitude calculated on the energy shell. Our numerical results for the angle-resolved HATI energy spectra show that the difference between the improved strong-field approximation and the low-frequency approximation is significant for scattering away from the laser polarization axis. In the context of quantum-orbit theory and the uniform approximation, we also show that on the back-rescattering ridge, the rescattering $T$-matrix element can be factorized into the product of the incoming flux and the elastic-scattering cross section so that the latter can be extracted from the angle- and energy-resolved HATI spectra.

Figure 1

(Color online) Two possible geometries of the (re)scattering event. For negative values of $k_s+A_s$ (upper panel) the scattering and laboratory systems have opposite $z$ axis so that ${\theta }_s=\phi -\pi$. For the lower panel these systems overlap so that ${\theta }_s=\phi$.

Figure 2

(Color online) Maximal electron kinetic energy $E_s$ at the time of rescattering (left ordinate—black solid line) and the corresponding rescattering time ${\tau }_s^{\prime }$ (right-hand ordinate—red dashed line) as functions of the scattering angle $\theta$. HATI of Xe atoms by the laser field having the intensity $1.5\times {10}^{14}\text{W}/{\text{cm}}^2$ and the wavelength 760 nm is considered.

Figure 3

(Color online) Maximal photoelectron momenta presented in the $\left(p_z,p_x\right)$ plane. Blue (solid) line: results for the drift momentum $\mathbf{p}$ obtained using the saddle-point method. Green line: vector potential ${\mathbf{A}}_r$ at the rescattering time which corresponds to the maximal drift electron energy for scattering angle $\theta =0$. Red dashed line (semicircle) is determined by the vector ${\mathbf{p}}_r$, defined in the text.

Figure 4

(Color online) Ionization rates of Ar presented in false colors in the $\left(p_z,p_x\right)$ momentum plane. The laser intensity and wavelength are $2.3\times {10}^{14}\text{W}/{\text{cm}}^2$ and 800 nm, respectively. The upper (lower) four panels are obtained using the first Born approximation (low-frequency approximation) for the rescattering matrix element. Each panel is denoted by the maximum number $\beta m$ used in the sum over the saddle-point solutions in the uniform approximation.

Figure 5

(Color online) Ionization rates of Ar presented in false colors in the electron emission angle-energy plane. The final electron energy is in units of the ponderomotive energy $U_p$. The laser parameters are as in Fig. 4. Ten pairs of quantum orbits are taken into account. The left (right) panel corresponds to the results obtained using the first Born approximation (low-frequency approximation) for the rescattering matrix element.

Figure 6

(Color online) Angular distributions of the differential cross sections for elastic scattering of electrons on Ar ions (black solid curves). The electron kinetic energies given in the various panels correspond to the incident energies $E_s$ which, after the electron rescattering and propagation in the laser field, lead to the final electron energy at the detector equal to the cutoff energy of the corresponding pair $\beta m$ of quantum orbits. Red dashed curves correspond to the results extracted from the HATI spectra of Fig. 4 (lowest right panel, LFA with ten pairs of orbits) for the same laser parameters as in Fig. 4.

Figure 7

(Color online) Angular distributions of the differential cross sections for elastic scattering of electrons on Ar ions (black solid curves). The electron kinetic energies used in the various panels correspond to the incident energies $E_s$ which, after the electron rescattering and propagation in the laser field, lead to the final electron energy at the detector equal to the cutoff energy of the denoted pair $\beta m$ of quantum orbits. Red dashed curves correspond to the results extracted from the HATI spectra obtained using LFA with nine pairs of orbits. The laser wavelength is 800 nm and the intensity $I$ is 1, 1.5, 3, and 3.5 in units of ${10}^{14}\text{W}/{\text{cm}}^2$, as denoted in each panel.

Figure 8

(Color online) Same as in Fig. 6 except that the HATI spectra, from which the data presented by the red dashed curve were extracted, were calculated taking into account only the pair $\beta m$ of quantum orbits, as explained in the text.

Figure 9

(Color online) Angular distributions of the differential cross section [black solid curve $\sigma \left(p_s,{\theta }_s\right)$] and the phase of the scattering amplitude [blue solid curve $\text{arg}f\left(p_s,{\theta }_s\right)$] for elastic scattering of electrons on Ar ions. Red dashed curve and green dot-dashed curve correspond to the cross section and the phase extracted from the HATI spectra along the BRR for the same laser parameters as in Fig. 4.

Figure 10

(Color online) Angular distributions of differential cross sections for elastic scattering of electrons on Ne, Ar, and Xe ions. The electron kinetic energies are 39, 23, and 16.5 eV for Ne, Ar, and Xe ions, respectively.