Towards the complete phase profiling of attosecond wave packets

Realistic attosecond wave packets have complex profiles that, in dispersive conditions, rapidly broaden or split in multiple components. Such behaviors are encoded in sharp features of the wave packet spectral phase. Here we exploit the quantum beating between one- and two-photon transitions in an attosecond photoionization experiment to measure the photoelectron spectral phase continuously across a broad energy range. Supported by numerical simulations, we demonstrate that this experimental technique is able to reconstruct sharp fine-scale features of the spectral phase, continuously as a function of energy and across the full spectral range of an attosecond pulse train. In a proof-of-principle experiment, we observe the periodic modulations of the spectral phase of an attosecond pulse train due to the individual chirp of each harmonic.

Figure 1

(a) RABBITT (2-2 quantum beat) and the 1-2 quantum beat protocol, schematically. Blue arrows indicate photoionization induced by the XUV, and red arrows indicate cc-transitions induced by the IR. (b) Amplitude and phase of the input XUV pulse. (c) RABBITT trace (total yield). The inset indicates the integration over all emission angles. (d) Asymmetry trace extracted from the 1-2 quantum beat. The inset indicates the asymmetry of the angular distribution (difference left-right). (e) Phase difference $\mathrm{\Delta }{\phi }_{2-2}\left(E\right)$ extracted from the RABBITT sidebands. (f) Phase differences $\mathrm{\Delta }{\phi }_{1-2}^{+}\left(E\right)$ and $\mathrm{\Delta }{\phi }_{1-2}^{-}\left(E\right)$ extracted from the 1-2 quantum beat method and comparison with $\mathrm{\Delta }{\phi }_{1-2}^{+,\mathrm{inp}.}\left(E\right)$ from the input phase.

Figure 2

(a) XUV-only (green) and delay-integrated RABBITT spectrum (blue) from the experiment. The red curve results from the fit of the transition ratios [Eq. (7)] to the integrated RABBITT spectrum in the range from 3 eV to 12 eV. (b) Measured asymmetry signal, defined as in Eq. (3), as a function of pump-probe delay. (c). Retrieved phases $\mathrm{\Delta }{\phi }_{1-2}^{+}\left(E\right)$ and $\mathrm{\Delta }{\phi }_{1-2}^{-}(E-\hslash {\omega }_{IR})$ from the experiment.

Figure 3

(a) XUV-only (green) and delay-integrated RABBITT spectrum (blue) from the quantum simulation employing a single-active electron (SAE) approximation and a model potential from Ref. [48]. The red curve results from a fit of the transition ratios [Eq. (7)] to the integrated RABBITT spectrum in the range from 2 eV to 14 eV. (b) Calculated asymmetry signal, defined as in Eq. (3), as a function of pump-probe delay. (c) Retrieved phases $\mathrm{\Delta }{\phi }_{1-2}^{+}\left(E\right)$ (red) and $\mathrm{\Delta }{\phi }_{1-2}^{-}(E-\hslash {\omega }_{IR})$ (blue) in comparison to $\mathrm{\Delta }{\phi }_{1-2}^{+,inp.}\left(E\right)$ from the input phase. The difference between the retrieved phase $\mathrm{\Delta }{\phi }_{1-2}^{+}\left(E\right)$ and the input phase $\mathrm{\Delta }{\phi }_{1-2}^{+,inp.}\left(E\right)$ is shown explicitly (green).

Figure 4

Comparison of simulated attosecond pulse trains (APTs) with periodic oscillations of the spectral phase (blue) and with a flat phase (green). The spectrum is identical for both APTs. The insets show the spectrum and the corresponding phase for both APTs.

Figure 5

(a) Photoelectron yield from the SAE simulation for helium for the two-color case with an XUV energy of 35 eV and IR intensity of $2\times {10}^{11}\mathrm{W}$/${\mathrm{cm}}^2$. The side peaks (green) correspond to the two-photon transitions for additional absorption or emission of one IR photon.

Figure 6

Relative amplitudes for the IR induced two-photon transitions as a function of energy for additional absorption and emission of an IR photon and for different intensities of the IR field (${10}^{11}$ W/${\mathrm{cm}}^2$ and $2\times {10}^{11}$ W/${\mathrm{cm}}^2$). The relative amplitudes for the higher intensity case are $\sim \sqrt{2}$ higher compared to the weak intensity case as they scale with the strength of the electric field. The energy axis refers to the electron energy of the corresponding one-photon amplitude.