Complete characterization of attosecond photoelectron wave packets

Development of attosecond laser technology allows us to measure electron dynamics such as photoionization delays between different species or electronic states. In general, the measured photoionization phase is a mixture of the spectral phase of the extreme ultraviolet (XUV) pulse and the atomic phases inherent to the optical transitions. Hence, it is difficult to disentangle these phases independently. Here we separate these phases by using an XUV attosecond pulse train containing both even and odd harmonic orders, generated by an 800- and 400 nm laser pulse, in the presence of the infrared 800-nm pulse. We measure the photoelectron angular distributions as a function of two independently controlled delays, the XUV-IR and the 800–400 nm delays, with attosecond time resolution. We analyze the photoelectron angular distributions to determine the relative amplitudes and phases of each angular momentum component. Using an in situ technique, we determine the phases of the harmonic orders and thereby completely determine the atomic phases. Using the obtained atomic phases and amplitudes, we reconstruct the real and imaginary parts of the continuum wave functions associated with three individual photoionization pathways.

Figure 1

Experimental setup. The main panel shows the view from top. Side panel (a) shows how to generate the second harmonic and make two delays. The side panel (b) shows how to isolate the XUV pulse from the copropagating IR pulse and adjust the delay between the XUV and IR pulses.

Figure 2

Three-path quantum interferences. (a) Illustration of an energy diagram for three-path interferences in the photoionization process caused by an XUV and a dressing IR pulse. Two color transitions by harmonic 13 plus one IR photon (H13+IR) and harmonic 15 minus one IR photon (H15-IR) generates a photoelectron wave with p- or f- angular component in the ionization continuum. One-photon ionization by harmonic 14 (H14) generates a $s$- or $d$- photoelectron wave. By tuning the photon energy of the XUV pulse and the intensity of the dressing IR pulse, we selectively ionize the magnetic quantum number $m=0$. (b) An example of a measured VMI image that is produced by the two-photon ionization processes, H15+IR and H13-IR. (c) and (d) Examples of VMI images when one-photon ionization by H14 is added to the two-photon ionization process. The images are taken at two different XUV-IR delays ($T_{\mathrm{XUV}}$) separated by 1.33 fs. Because of the three-path interference, the angular distribution varies with $T_{\mathrm{XUV}}$. The polarization direction of both XUV and IR pulses is indicated by the black arrow.

Figure 3

In situ measurement of the harmonic phase. (a) The high-harmonic spectra in argon measured as a function of $T_{2\omega }$, the delay between the fundamental and second harmonic pulses. The white dotted lines indicate those delays at which VMI measurements were made. The time zero is set at the delay when the H14 intensity is maximized. The red dots mark those times at which the even harmonic intensity minimizes. (b) The emission times (relative to H14) obtained from the measurement (data points). The curve shows the classical calculation of the relative emission time for each harmonic order in argon at the 800 nm intensity of $1.2\times {10}^{14}\mathrm{W}/\mathrm{c}{\mathrm{m}}^2$.

Figure 4

Photoelectron angular distributions as a function of delays. The experimental (left column) and modeled (right column) photoelectron angular distributions as a function of the XUV-IR delay ($T_{\mathrm{XUV}}$) at 8 values of the delay between the 800 and 400 nm driving fields ($T_{2\omega }$). The values of $T_{2\omega }$ are shown on the left side in $fs$. The derived fitting parameters, amplitudes and phases of angular momentum components, are listed in Table 1.

Figure 5

Phases of photoelectron angular components as a function of $T_{2\omega }$. (a) The fitted phases ${\varphi }_{p13}^F,{\varphi }_{p15}^F,{\varphi }_{f13}^F$ and ${\varphi }_{f15}^F$ listed in Table 1. The data points at $T_{2\omega }=0.749fs$ are significantly shifted from neighboring points due to the fitting procedure (see text). (b) The phases averaged over ${\varphi }_{p13}^F,{\varphi }_{p15}^F,{\varphi }_{f13}^F$ and ${\varphi }_{f15}^F$. The phase at $T_{2\omega }=0$ is set to zero. The lengths of the error bars are normalized so that the error at $T_{2\omega }=0$ is zero. A π phase jump (3.23 radians) is seen at $T_{2\omega }\sim$ 0.33 fs. (c) The calculated spectral phases of H14 (red) and H18 (blue) versus $T_{2\omega }$. The SFA calculation is compared with the experimental data (square data points). The spectral phases for H13 (magenta) and H15 (green) are also plotted. (d) The phases after removing the $\pi$ phase jump. The dashed lines represent the phases averaged over $T_{2\omega }$ for each partial wave component.

Figure 6

Reconstruction of the photoelectron wave packets. The 3D and 2D re-constructed momentum-space electron wave-packet images produced by photoionization pathways of (a) H15−IR, (b) H13+IR, and (c) H14. The left panels show the real part and the right panels show the imaginary part of the wave packet. The 2D images are obtained by cutting the 3D images along the polarization axis, vertical in the figure. We use the atomic phases and amplitudes for each partial wave given in the right side of Table 2.

Figure 7

Scattering phase and continuum-continuum phase. Scattering phases ${\eta }_l$ of the continuum electron wave function for each partial wave component, calculated for neon, as a function of the photon energy. The first ionization potential of neon is 21.56 eV. Additional phase shift ${\varphi }_{cc}$ of the photoelectron wave function due to continuum-continuum coupling by the infrared field. The phase diverges quickly as the electron kinetic energy becomes small.