Analysis of two-dimensional photoelectron momentum spectra and the effect of the long-range Coulomb potential in single ionization of atoms by intense lasers

Two-dimensional (2D) electron momentum distributions and energy spectra for multiphoton ionization of atoms by intense laser pulses, calculated by solving the time-dependent Schrödinger equation (TDSE) for different wavelengths and intensities, are compared to those predicted by the strong-field approximation (SFA). It is shown that the momentum spectra at low energies between the TDSE and SFA are quite different and the differences arise largely from the absence of a long-range Coulomb interaction in the SFA. We further found that the low-energy 2D momentum spectra from the TDSE exhibit ubiquitous fanlike features where the number of stripes is due to a single dominant angular momentum of the low-energy electron. The specific dominant angular momentum in turn has been found to be decided by the minimum number of photons needed to ionize the atom only. The electron momentum spectra predicted by models modified from the SFA are also examined and found to lack the fanlike features as in the SFA.

Figure 1

(Color online) R-weighted potential, or effective charge, of argon, together with models with potentials cut off at different $r_c$ (see text).

Figure 2

(Color online) Photoelectron 2D momentum distributions for single ionization of argon by a $10\text{−}\mathrm{fs}$ (FWHM) laser pulse with wavelength of $400\mathrm{nm}$ at the peak intensity of $3.2\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, calculated by the (a) full TDSE, (b) TDSE with $r_c=5$, (c) TDSE with $r_c=2$, and (d) SFA.

Figure 3

(Color online) Photoelectron 2D momentum distributions for single ionization of argon by a $10\text{−}\mathrm{fs}$ (FWHM) laser pulse with wavelength of $600\mathrm{nm}$ at the peak intensity of $1.4\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, calculated by the (a) full TDSE, (b) TDSE with $r_c=5$, (c) TDSE with $r_c=2$, and (d) SFA.

Figure 4

(Color online) Same as Fig. 2 but for the low-energy region.

Figure 5

(Color online) (a) Absolute electron energy spectra of Fig. 2. (b) Same spectra but normalized at the first peak.

Figure 6

(Color online) (a) Perpendicular momentum distribution from the spectra of Fig. 2. (b) Same spectra but normalized at the peak.

Figure 7

(Color online) Photoelectron 2D momentum distributions for single ionization by a $10\text{−}\mathrm{fs}$ (FWHM) laser pulse. (a) Ar, with a wavelength of $400\mathrm{nm}$ at peak intensity of $1.7\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$; (b) Ar, with a wavelength of $590\mathrm{nm}$ at peak intensity of $2.8\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$; (c) Ne, with a wavelength of $600\mathrm{nm}$ at peak intensity of $2.1\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$; (d) H, with a wavelength of $800\mathrm{nm}$ at peak intensity of $0.8\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$; (e) H, with a wavelength of $800\mathrm{nm}$ at peak intensity of $2.03\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$; (f) He, with a wavelength of $400\mathrm{nm}$ at peak intensity of $6.2\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 8

(Color online) Dependence of the dominant $L$ on the minimum number $N$ of photons needed to ionize the atom for atoms initially in the $s$ or $p$ state.

Figure 9

(Color online) Photoelectron spectra for single ionization of argon by a $10\mathrm{fs}$ (FWHM) laser pulse with the wavelength of $400\mathrm{nm}$ at the peak intensity of $3.2\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. (a) 2D momentum distribution calculated by SFA, (b) 2D momentum distribution calculated by MSFA, and (c) the SFA and MSFA energy distribution.

Figure 10

(Color online) Comparison of photoelectron spectra of H $\left(1s\right)$ by a $10\text{−}\mathrm{fs}$ (FWHM) laser pulse with wavelength of $400\mathrm{nm}$ at peak intensity of (a) $1.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ and (b) $2.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, calculated by different theoretical models. Note that the results from the SFA have been normalized, as shown on the labels. See text.