Multistart spiral electron vortices in ionization by circularly polarized UV pulses

Multistart spiral vortex patterns are predicted for the electron momentum distributions in the polarization plane following ionization of the helium atom by two time-delayed circularly polarized ultrashort laser pulses. For two ultraviolet (UV) pulses having the same frequency (such that two photons are required for ionization), single-color two-photon interferometry with corotating or counter-rotating time-delayed pulses is found to lead respectively to zero-start or four-start spiral vortex patterns in the ionized electron momentum distributions in the polarization plane. In contrast, two-color one-photon plus two-photon interferometry with time-delayed corotating or counter-rotating UV pulses is found to lead respectively to one-start or three-start spiral vortex patterns. These predicted multistart electron vortex patterns are found to be sensitive to the carrier frequencies, handedness, time delay, and relative phase of the two pulses. Our numerical predictions are obtained by solving the six-dimensional two-electron time-dependent Schrödinger equation (TDSE). They are explained analytically using perturbation theory (PT). Comparison of our TDSE and PT results for single-color two-photon processes probes the role played by the time-delay-dependent ionization cross channels in which one photon is absorbed from each pulse. Control of these cross channels by means of the parameters of the fields and the ionized electron detection geometries is discussed.

Figure 1

Schematic energy-level diagrams for the cases of single ionization of He (with binding energy $E_b=24.6$ eV) by two circularly polarized pulses delayed in time by $\tau$ that are considered in this paper: (a) Two-photon interferometry for a carrier frequency ${\omega }_1=15$ eV, which leads to either zero-start (0-S) or four-start (4-S) vortex patterns for equal or opposite helicity pulses; and (b) two-color interferometry for carrier frequencies ${\omega }_2=2{\omega }_1=30$ eV, which leads to either one-start (1-S) or three-start (3-S) vortex patterns for equal or opposite helicity pulses. In each case, the ionization pathways ${\mathrm{\Gamma }}_1$, ${\mathrm{\Gamma }}_{12}$ and/or ${\mathrm{\Gamma }}_2$ are shown.

Figure 2

Triply differential probability (TDP) $d^{3}W/d^3\mathbf{p}$ [see Eqs. (5), (23), (8), and (20)] in the polarization plane for single ionization of He by two right circularly polarized pulses delayed in time by (a) $\tau =0$, and (b) $\tau =1.65$ fs. Each cosine-squared pulse has a carrier frequency ${\omega }_1=15$ eV, $N=6$ optical cycles, total duration $T=1.65$ fs, and peak intensity $I={10}^{12}\text{W}/{\mathrm{cm}}^2$. The CEPs are ${\varphi }_1=0$ and ${\varphi }_2=0$. The magnitudes of the TDPs (in units of ${10}^{-7}$ a.u.) are indicated by the color scales in (a) and (b).

Figure 3

Energy distributions in the polarization plane for electrons ejected at the angle $\phi =0$ following two-photon single ionization of He by two right circularly polarized pulses delayed in time by (a) $\tau =0$, (b) $\tau =500$ as, (c) $\tau =800$ as, (d) $\tau =1.102$ fs, (e) $\tau =1.378$ fs, and (f) $\tau =1.654$ fs. Results are obtained by ab initio TDSE calculations using Eq. (5). For comparison, we also show results using PT formula (24) [see panel (a)] or (25) [see panels (b)–(f)], in which the dynamical parameter ${\mathcal{M}}_d\left(p\right)$ is extracted numerically from ab initio TDSE calculations for a single pulse including only the $L=2,M_L=+2$ amplitude; see Appendix pp2. The pulse parameters are the same as in Fig. 2.

Figure 4

Four-start spiral vortex patterns in the photoelectron momentum distributions $d^{3}W/d^3\mathbf{p}$ [see Eq. (5)] in the polarization plane following ionization of helium by right and left circularly polarized pulses delayed in time by (a) and (b) $\tau =0$, (c) $\tau =500$ as, (d) $\tau =800$ as, (e) $\tau =1.1$ fs, and (f) $\tau =1.65$ fs. In (a) and (c)–(f), ${\varphi }_1={\varphi }_2=0$; in (b), ${\varphi }_1=0$, ${\varphi }_2=\pi /2$. Each pulse has ${\omega }_1=15$ eV, $N=6$ cycles, and $I={10}^{12}\text{W}/{\mathrm{cm}}^2$. The magnitudes of the TDPs (in units of ${10}^{-7}$ a.u.) are indicated by the color scales. In all panels, the numerical projections (5) include $L=0,2$ with $M_L=0,\pm 2$.

Figure 5

Same as Fig. 4 but the numerical results are obtained using the PT formulas (31) and (20) with the dynamical parameter ${\mathcal{M}}_d\left(p\right)$ extracted numerically from ab initio TDSE calculations for a single pulse including only $L=2,M_L=\pm 2$ amplitudes (see Appendix pp2). Note that Eq. (31) excludes cross channel contributions.

Figure 6

Energy distributions in the polarization plane for electrons detected at the angle $\phi =0$ following two-photon single ionization of He by two oppositely circularly polarized pulses delayed in time by (a) $\tau =0$ as, (b) $\tau =500$ as, (c) $\tau =800$ as, and (d) $\tau =1.1$ fs. Parameters of the two pulses are the same as in Fig. 4. Results are obtained from TDSE calculations for the following final states: (i) $L=0,2$ with $M_L=0,\pm 2$ (solid black lines), (ii) $L=2$ with $M_L=\pm 2$ (dashed red lines); and from the formula (31), in which the dynamical parameter ${\mathcal{M}}_d\left(p\right)$ is extracted numerically (as described in Appendix pp2) from an ab initio TDSE calculation for a single pulse including only $L=2,M_L=\pm 2$ amplitudes (dash-dotted blue lines).

Figure 7

Angular distributions $d^{3}W/d^3\mathbf{p}$ (in units of ${10}^{-7}$ a.u.) in the polarization plane for a fixed ionized electron energy of $E=2{\omega }_1-E_b$ following two-photon single ionization of He by two oppositely circularly polarized pulses delayed in time by (a) $\tau =0$ as, (b) $\tau =500$ as, (c) $\tau =800$ as, and (d) $\tau =1.1$ fs. Parameters of the two pulses are the same as in Fig. 4. Results are obtained from TDSE calculations for the following final states: (i) $L=0,2$ with $M_L=0,\pm 2$ (solid black lines), (ii) $L=2$ with $M_L=\pm 2$ (dashed red lines); and from the formula (31) as described in the caption of Fig. 6 (dash-dotted blue lines).

Figure 8

Time-delay periodicity of electron angular distributions $d^{3}W/d^3\mathbf{p}$ (in units of ${10}^{-7}$ a.u.) in the polarization plane produced by two-photon single ionization of the He ground state by two oppositely circularly polarized UV pulses. The ionized-electron energy is fixed to be $E=2{\omega }_1-E_b$. Parameters of the two pulses are the same as in Fig. 4. Results in panel (a) are obtained by ab initio TDSE calculations for $L=0,2$ with $M_L=0,\pm 2$ final states; results in panel (b) are obtained using PT formula (31) calculated as described in the caption of Fig. 6. In each panel, results are shown for three time delays: ${\tau }_8$, ${\tau }_{12}$, and ${\tau }_{14.5}$, where ${\tau }_n=n\pi /{\omega }_1$.

Figure 9

Cross channel path (${\mathrm{\Gamma }}_{12}$) effects for two time-delayed linearly polarized pulses. (a) Angular distributions $d^{3}W/d^3\mathbf{p}$ (in units of ${10}^{-3}$ a.u.) in the ($p_z,p_x$) plane for two-photon single ionization of the He ground state by two time-delayed orthogonal pulses linearly polarized along the $x$ and $y$ axes, respectively. Results for time delays ${\tau }_0$ and ${\tau }_6$ are shown, where ${\tau }_n=n\pi /{\omega }_1$. (b) Angular distributions $d^{3}W/d^3\mathbf{p}$ (in units of ${10}^{-2}$ a.u.) in the ($p_x,p_y$) plane for two-photon single ionization of the He ground state by two time-delayed pulses linearly polarized along the $x$ axis. Results for four time delays are shown, namely, ${\tau }_0$, ${\tau }_6$, ${\tau }_8$, and ${\tau }_{12}$. In both (a) and (b), the ionized electron energy is $E=2{\omega }_1-E_b$; each ${\text{cos}}^2$ pulse has a carrier frequency ${\omega }_1=15$ eV, $N=6$ cycles, an intensity of $I=5\times {10}^{13}\text{W}/{\mathrm{cm}}^2$, and a zero CEP.

Figure 10

One-start spiral vortex patterns in the electron momentum distribution $d^{3}W/d^3\mathbf{p}$ [see Eq. (5)] in the polarization plane following one-photon or two-photon single ionization of He by two identical circularly polarized pulses for three time delays: (a) $\tau =0$, (b),(d) $\tau =500$ as, and (c) $\tau =1$ fs. In (a)–(c), the two pulses are both right circularly polarized, whereas in (d) they are both left circularly polarized. The carrier frequency and intensity of the earlier (later) pulse is ${\omega }_2=30$ eV (${\omega }_1=15$ eV) and $I_2=10{\mathrm{TW}\mathrm{cm}}^{-2}\left(I_1=100{\mathrm{TW}\mathrm{cm}}^{-2}\right)$. The magnitudes of the TDPs (in units of ${10}^{-3}$ a.u.) are indicated by the color scales.

Figure 11

Three-start spiral vortex patterns in the electron momentum distribution $d^{3}W/d^3\mathbf{p}$ [see Eq. (5)] in the polarization plane following one-photon or two-photon single ionization of He by two oppositely circularly polarized pulses for three time delays: (a) $\tau =0$, (b),(d) $\tau =500$ as, and (c) $\tau =1$ fs. In (a)–(c), pulses are right and left circularly polarized, whereas in (d) they are left and right circularly polarized. The pulse parameters are the same as in Fig. 10. The magnitudes of the TDPs (in units of ${10}^{-3}$ a.u.) are indicated by the color scales.