Nondipole effects in atomic dynamic interference

Nondipole effects in the atomic dynamic interference are investigated by numerically solving the time-dependent Schrödinger equation (TDSE) of hydrogen. It is found that the inclusion of nondipole corrections in the TDSE can induce momentum shifts of photoelectrons in the opposite direction of the laser propagation. The magnitude of the momentum shift is roughly proportional to the laser peak intensity and to the momentum component of the photoelectron along the laser propagation. By including the nondipole corrections of the Volkov phase into a semianalytical model previously developed under the dipole approximation, all the main features of the momentum shifts can be nicely reproduced. Through an analytic expression, the origin of such momentum shifts is attributed to the nondipole phase difference between the two electron wave packets ejected in the rising edge and the falling edge of the laser pulse, which will interfere with each other and result in the final fringe pattern. One important consequence of such momentum shifts is that they can smooth out the peak splitting induced by the dynamic interference in the photoelectron energy spectrum. Nevertheless, it should be emphasized that the dynamic interference persists in the photoelectron momentum distributions and is not suppressed at all for the laser parameters considered in this work.

Figure 1

The photoelectron energy distribution $D\left(E\right)$ for the $1s$ state of hydrogen exposed to linearly polarized Gaussian-shaped pulses with a carrier frequency of $\omega =53.6$ eV and FWHM of 7 cycles at four different laser intensities: (a) $I_0=1.0\times {10}^{17}\mathrm{W}/{\mathrm{cm}}^2$, (b) $I_0=1.0\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$, (c) $I_0=4.0\times {10}^{18}\mathrm{W}/{\mathrm{cm}}^2$, and (d) $I_0=1.0\times {10}^{19}\mathrm{W}/{\mathrm{cm}}^2$. Red solid line: the dipole result; blue dotted line: the nondipole result.

Figure 2

The photoelectron momentum distribution $D\left(\mathbf{p}\right)$ in the $p_x\text{−}{p}_z$ plane ($p_y=0$) at $I_0=1.0\times {10}^{19}\mathrm{W}/{\mathrm{cm}}^2$. The upper (lower) half plane is the dipole (nondipole) result with a black solid (dashed) semicircle as a reference to highlight the momentum shift.

Figure 3

(a) The momentum shift $\mathrm{\Delta }p$ at various laser intensities: results in red solid circles (green open circles) are values extracted from the momentum distributions calculated by the nondipole TDSE (the semianalytical model); the blue crosses refer to the analytic results from Eq. (13). (b) Angle-integrated energy spectra $D\left(E\right)$: for the dipole result as a red dotted line, and for the nondipole result respectively calculated before (after) moving the center ${\mathrm{O}}^{\prime }$ back to zero by a blue dashed (green solid) line.

Figure 4

The two-dimensional (2D) photoelectron momentum distributions $D\left(\mathbf{p}\right)$ in the $p_x\text{−}{p}_z$ plane ($p_y=0$) at $I_0=1.0\times {10}^{19}\mathrm{W}/{\mathrm{cm}}^2$ for (a) the dipole TDSE calculation, (b) the nondipole TDSE calculation for $p_x>0$, (c) the nondipole TDSE calculation for $p_x<0$, and (d) the semianalytical model results for $p_x>0$.

Figure 5

The upper row shows the 2D photoelectron momentum distribution $D\left(\mathbf{p}\right)$ in the $p_y\text{−}{p}_z$ plane ($p_x=0$) at $I_0=1.0\times {10}^{19}\mathrm{W}/{\mathrm{cm}}^2$: (a) the dipole TDSE calculation for $p_y>0$, and (b) the nondipole TDSE calculation for $p_y>0$. The lower row shows both the dipole and the nondipole $D\left(p\right)$ results for photoelectrons with $p_x=0$: (c) along the laser polarization direction, and (d) along the direction of $\theta =\pi /4$.