Analysis of two-dimensional high-energy photoelectron momentum distributions in the single ionization of atoms by intense laser pulses

We analyzed the two-dimensional (2D) electron momentum distributions of high-energy photoelectrons of atoms in an intense laser field using the second-order strong field approximation (SFA2). The SFA2 accounts for the rescattering of the returning electron with the target ion to first order and its validity is established by comparing with results obtained by solving the time-dependent Schrödinger equation for short pulses. By analyzing the SFA2 theory, we confirmed that the yield along the back rescattered ridge in the 2D momentum spectra can be interpreted as due to the elastic scattering in the backward directions by the returning electron wave packet. The characteristics of the extracted electron wave packets for different laser parameters are analyzed, including their dependence on the laser intensity and pulse duration. For long pulses we also studied the wave packets from the first and the later returns.

Figure 1

(Color online) Electron energy spectra for atomic hydrogen by a five-cycle laser pulse with the wavelength of $800\mathrm{nm}$ at the peak intensity of $1.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. (a) From TDSE, first- (SFA1) and second-order (SFA2) theory and the coherent sum of SFA1 and SFA2; (b) the sum of SFA1 and SFA2 but renormalized to the high-energy part of TDSE. Electron energy is in units of $U_p$, the ponderomotive energy.

Figure 2

(Color online) Photoelectron 2D momentum distributions parallel $(p_{\Vert }=p_z)$ and perpendicular $\left(p_y\right)$ to the laser polarization direction for atomic hydrogen by a five-cycle laser pulse with the wavelength of $800\mathrm{nm}$ at the peak intensity of $1.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$. (a) SFA2 and (b) TDSE.

Figure 3

(Color online) Photoelectron 2D momentum distributions parallel $(p_{\Vert }=p_z)$ and perpendicular $(p_{\perp }=\sqrt{p_x^2+p_y^2})$ to the laser polarization direction for atomic hydrogen by laser pulses at the peak intensity of $1.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ with the wavelength of $800\mathrm{nm}$ for durations of two cycles and three cycles, respectively. The corresponding laser fields and vector potentials are shown on the right for the analysis of the rescattering mechanism.

Figure 4

(Color online) Analysis of 2D momentum distributions as in Fig. 3 for the three cycle pulse, for the identification of the tunneling time and rescattering time, for the right- and left-side BRR electrons. See text.

Figure 5

(Color online) Angle-resolved energy spectra for atomic hydrogen by a five-cycle laser pulse with the wavelength of $800\mathrm{nm}$ at the peak intensity of $1.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, for four detector angles of $\theta =0°$, 11.2°, 22.5°, and 34°, respectively. The arrows indicate the energies predicted by Eq. (28), see text for detail.

Figure 6

(Color online) 2D momentum distributions for atomic hydrogen by five-cycle laser pulses with the same Keldysh parameter $\gamma =1.07$, but for wavelengths of 400, 600, 800, and $1000\mathrm{nm}$, respectively.

Figure 7

(Color online) Right-side BRR for atomic hydrogen by a five-cycle laser pulse at the peak intensity of $1.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ with the wavelength of $800\mathrm{nm}$. (a) Comparison of the first Born elastic scattering cross sections with the angular distributions on BRR calculated from SFA2. (b) Comparison of the electron wave packet from SFA2 with that from TDSE at ${\theta }_r=160°$. (c) The wave packets of the returning electrons are shown to be independent of the scattering angles.

Figure 8

(Color online) Distributions of the momentum of the right-side BRR electrons for atomic hydrogen by five-cycle laser pulses with the wavelength of $800\mathrm{nm}$ at the peak intensities of (a) $5.0\times {10}^{13}$, (b) $1.0\times {10}^{14}$, (c) $2.5\times {10}^{14}$, and (d) $5.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$.

Figure 9

(Color online) Ionization of hydrogen by laser pulses at the peak intensity of $1.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ with the wavelength of $800\mathrm{nm}$ and the full durations of 10 and $20\text{cycles}$. (a) and (b) Comparison of the first Born elastic scattering cross sections with the SFA2 angular distributions of the BRR electrons. (c) and (d) Distributions of $I(p_r,{\theta }_r)$ at ${\theta }_r=140°$ and 180°.

Figure 10

(Color online) Distributions of $I(p_r,{\theta }_r)$ of the BRR electrons, born at the time $(-0.5\tau ,-0.4\tau )$ and returning at the first-return time $(0.0,0.5\tau )$, the second-return time $(0.5\tau ,1.0\tau )$ and the third-return time $(1.0\tau ,1.5\tau )$, at ${\theta }_r=170°$ for ionization of hydrogen by a $25\text{−}\text{cycle}$ laser pulse at the peak intensity of $1.0\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ with the wavelength of $800\mathrm{nm}$.