Electron Vortices in Photoionization by Circularly Polarized Attosecond Pulses

Single ionization of He by two oppositely circularly polarized, time-delayed attosecond pulses is shown to produce photoelectron momentum distributions in the polarization plane having helical vortex structures sensitive to the time delay between the pulses, their relative phase, and their handedness. Results are obtained by both ab initio numerical solution of the two-electron time-dependent Schrödinger equation and by a lowest-order perturbation theory analysis. The energy, bandwidth, and temporal duration of attosecond pulses are ideal for observing these vortex patterns.

Figure 1

Triply differential probability (TDP) $d^{3}W/d^3\mathbf{p}$ [see Eqs. (2), (3), and (4)] in the polarization plane for ionization of He by two right-circularly polarized pulses delayed in time by (a) $\tau =0$; and (b) $\tau =500\mathrm{as}$. Each pulse has carrier frequency $\omega =36\mathrm{eV}$, $n_p=3$ cycles with total duration $T=344\mathrm{as}$, intensity $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, and CEPs of ${\varphi }_1=0$ and ${\varphi }_2=\pi /2$. The magnitudes in a.u. of the TDPs are indicated by the color scales in each panel.

Figure 2

Triply differential probability (TDP) $d^{3}W/d^3\mathbf{p}$ [see Eqs. (2), (4), and (5)] in the polarization plane for He ionization by (a),(b),(c) right-left and (d) left-right circularly polarized attosecond pulses. Top row results: time delay $\tau =0$; bottom row results: $\tau =500\mathrm{as}$. In (a), ${\varphi }_1={\varphi }_2=0$; in (b), (c), (d), ${\varphi }_1=0$, ${\varphi }_2=\pi /2$. In all panels $\omega =36\mathrm{eV}$, $n_p=3$ cycles, $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, and the magnitudes in a.u. of the TDPs are indicated by the color scales.

Figure 3

Photoelectron momentum distributions $d^{3}W/d^3\mathbf{p}$ [see Eqs. (2), (4), and (5)] in the polarization plane for ionization of He by right-left circularly polarized attosecond pulses delayed in time by (a) $\tau =50\mathrm{as}$; (b) $\tau =150\mathrm{as}$; (c) $\tau =373\mathrm{as}$; and (d) $\tau =976\mathrm{as}$. Pulse parameters are the same as in Figs. 2 and 2. The magnitudes in a.u. of the TDPs are indicated by the color scales.

Figure 4

Angular distributions $d^{3}W/d^3\mathbf{p}$ (in units of ${10}^{-3}\mathrm{a}.\mathrm{u}.$) for a fixed photoelectron energy $E_1=\omega -E_b$, produced by a pair of right-left circularly polarized attosecond pulses. Results in panel (a) are obtained by ab initio TDSE calculations [see Eq. (2)]; results in panel (b) are obtained using the PT formula (5) in which the dynamical parameter $W_p$ is determined from our TDSE results for a single right-circularly polarized attosecond pulse including only the ${{}^{1}P}^o$ final state amplitude. In each panel, results are shown for three time delays: ${\tau }_0$, ${\tau }_{10}$, and ${\tau }_{17}$, where ${\tau }_n=n\pi /\omega$.