Two-color polarization control of angularly resolved attosecond time delays

Measured photoionization time delays may exhibit large variations as a function of the emission angles, even for spherically symmetric targets, as shown in recent reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) experiments. The contributions from different pathways to the two-photon quantum channels can explain the observed phase jumps that shape those angular distributions. Here we propose a simple analytical model to describe angularly resolved RABBITT spectra as a function of the relative polarization angle between the ionizing attosecond pulse train and the assisting IR field. We demonstrate that the angular dependences of the measured delays can be analytically predicted and the position of the phase jumps reduced to the analysis of a few relevant parameters.

Figure 1

(a) Schematic representation of pathways contributing to consecutive main bands and the sideband in between. For low IR intensities, main bands are mainly populated by transitions from the ground state to continuum states, triggered by the absorption of a single photon with energy ${\mathrm{\Omega }}_{\pm }=(2q\pm 1){\omega }_0$, from the attosecond pulse train. The interaction of these primary photoelectron distributions with the IR gives rise to the sidebands. The reverse contributions, where the IR photon is absorbed or emitted first, are not shown here. (b) Geometric configuration. We indicate the polarization direction of the XUV laser source by the blue vector ${\widehat{\varepsilon }}_{\text{XUV}}$ collinear to the Cartesian $x$ axis. The polarization vector of the IR laser ${\widehat{\varepsilon }}_{\text{IR}}$ lies in the $xy$ plane and subtends an angle $\mathrm{\Theta }$ with the Cartesian $x$ axis. The final photoelectron momentum also lies in the $xy$ plane and it is indicated by the green vector ${\mathbf{k}}_e$.

Figure 2

Comparison of the angular variation of time delays at photoelectron energies coinciding with RABBITT sidebands for hydrogen and helium atoms. Theoretical calculations for the hydrogen atom are computed through three different methods: TDSE, SOPT, and ACC-RME calculations. The data from Refs. [17, 27] correspond to the helium atom. The photoelectron kinetic energy of final states in both systems are indicated in each panel. The sideband index corresponds to hydrogen. The phase jump angle ${\theta }_j^{+}$ is calculated from Eq. (11) and the relative time delay for that emission angle is obtained from linear interpolation from the TDSE results. This leads to larger fluctuations for sidebands with larger index due to the steeper variations of $\mathrm{\Delta }{\tau }_{\text{tot}}$. The inset in figure for ${\mathrm{SB}}_{12}$ shows the ratios $|T_0^{\pm }/T_2^{\pm }|$ obtained from SOPT (stars) and ACC-RME (lines) calculations [33]. The other insets show the close-up image of relative time delays for the gray-shaded areas.

Figure 3

Angularly resolved atomic time delays at photoelectron energies coinciding with RABBITT sidebands for hydrogen atoms and different relative polarization angles $\mathrm{\Theta }$. The results are computed through three different methods: TDSE, SOPT, and ACC-RME calculations. Theoretical and experimental data for helium were taken from Ref. [27]. The phase jump angles ${\theta }_j^{\pm }$ were obtained from Eq. (12) and the relative time delay for that emission angle was obtained from linear interpolation of the TDSE results. This procedure may lead to larger fluctuations for sidebands with larger index due to the steeper variation of $\mathrm{\Delta }{\tau }_{\text{tot}}$. We display only the first occurrence of the phase jump angles ${\theta }_j^{\pm }$ associated with absorption and emission pathways. The second phase jump angle for each family arises at emission angles symmetric with respect to $(\pi +\mathrm{\Theta })/2$.

Figure 4

Time-delay-independent photoelectron emission probabilities $[A$ term in Eq. (1)] as a function of photoelectron emission angle $\theta$ for different relative polarization angles $\mathrm{\Theta }$ between the XUV and the IR fields. Theoretical and experimental data for helium were taken from Ref. [27].