Complete characterization of single-cycle double ionization of argon from the nonsequential to the sequential ionization regime

Selected features of nonsequential double ionization have been qualitatively reproduced by a multitude of different (quantum and classical) approaches. In general, however, the typical uncertainty of laser pulse parameters and the restricted number of observables measured in individual experiments leave room for adjusting theoretical results to match the experimental data. While this has been hampering the assessment of different theoretical approaches leading to conflicting interpretations, comprehensive experimental data that would allow such an ultimate and quantitative assessment have been missing so far. To remedy this situation we have performed a kinematically complete measurement of single-cycle multiple ionization of argon over a one order of magnitude range of intensity. The momenta of electrons and ions resulting from the ionization of the target gas are measured in coincidence, while each ionization event is tagged with the carrier-envelope phase and intensity of the 4-fs laser pulse driving the process. The acquired highly differential experimental data provide a benchmark for a rigorous test of the many competing theoretical models used to describe nonsequential double ionization.

Figure 1

Ratio of the recorded yields of ${\mathrm{Ar}}^{2+}$ and ${\mathrm{Ar}}^{+}$ ions plotted as a function of intensity for both data sets (light and dark blue, respectively). The measured TEMS are CEP averaged and contain events collected in a $\pm 5\%$ intensity interval around the value indicated in units of ${10}^{14}\text{W}{\text{cm}}^{-2}$ in each panel. The dashed blue line is the calculated intensity-dependent yield ratio expected from SDI.

Figure 2

Calculated CEP-averaged TEMS for double ionization of Ar assuming (a) the RESD model for NSDI and (b) SDI. The number in the top left corner of each TEMS indicates the scaled intensity $I_{\mathrm{th}}$, in units of ${10}^{14}\text{W}{\text{cm}}^{-2}$, used in the calculation. The TEMS in (a) are normalized to their respective maxima, while the spectra in (b) are normalized to the maximum signal at the highest intensity shown. For visibility, the SDI signal calculated for the two lowest intensity values is enhanced by a factor of 500 or 10, respectively. The value of $\beta$, the only free parameter of the RESD model (see Appendix pp1), is indicated in each panel.

Figure 3

Comparison of the measured intensity-dependent ratio of double to single ionization yields (markers) to the calculation results (lines). The dashed and solid lines represent the calculated SDI and NSDI contribution, respectively. The sum of these contributions is plotted as a dotted line. The set of blue curves on the right and the set of green curves on the left have been calculated using the measured $I$ and scaled intensities $I_{\text{th}}$, respectively.

Figure 4

Intensity dependence of the CEP-dependent asymmetries in the ${\mathrm{Ar}}^{+}$ and ${\mathrm{Ar}}^{2+}$ ion yields, encoded as blue and black, respectively. The CEP-dependent asymmetry is parametrized according to Eq. (2) and the intensity dependences of $A_0\left(I\right)$ and ${\varphi }_0\left(I\right)$ are shown in (a) and (b), respectively. The data recorded in the low-intensity and high-intensity measurements are represented by circles and crosses, respectively. The blue solid line corresponds to the SI calculations. The solid and dashed black lines display the predictions by the semiclassical NSDI and SDI models, respectively. Here the unknown phase offset is chosen such that the values of ${\varphi }_0\left(I\right)$ recorded for ${\mathrm{Ar}}^{+}$ at $I_{\mathrm{th}}=3\times {10}^{14}\text{W}{\text{cm}}^{-2}$ agree with the result of the SI calculation.

Figure 5

(a)–(f) CEP-averged ${\mathrm{Ar}}^{+}$ momentum spectra along the laser polarization (blue symbols) measured at various intensities, shown together with the simulated spectra (solid green lines) calculated at a scaled intensity of 2.5 times the experimental intensity. The black and green numbers in each panel indicate the measured and scaled intensity in units of ${10}^{14}\text{W}{\text{cm}}^{-2}$ for the experiment and theory, respectively.

Figure 6

Same as Fig. 1 but showing the CEP-averaged ${\mathrm{Ar}}^{2+}$ ion momentum spectra along the laser polarization instead of the TEMS. The measured spectra (black symbols) are shown together with the calculated ion momentum spectra assuming a weighted sum (solid green line) of RESD (solid line) and SDI (dashed line) contributions. The ratio $r$ of SDI to the total signal is indicated in each panel. The lower intensity $I$ axis denotes the measured intensity and the upper intensity axis $I_{\mathrm{th}}$ denotes the scaled intensity used in the calculations. The black and green numbers in each panel indicate, in units of ${10}^{14}\text{W}{\text{cm}}^{-2}$, the measured and scaled intensities, respectively. All spectra are normalized to a peak value of 1. The red curve displays the yield ratio of ${\mathrm{Ar}}^{3+}$ to ${\mathrm{Ar}}^{+}$. Also shown is the corresponding CEP- and intensity-averaged ${\mathrm{Ar}}^{3+}$ ion momentum spectrum.

Figure 7

Intensity dependence of the CEP-dependent asymmetries in the ${\mathrm{Ar}}^{n+}$ ($n=1,2,3$) ion yields, encoded as blue, black, and red, respectively. The intensity dependences of $A_0\left(I\right)$ and ${\varphi }_0\left(I\right)$ are shown in (a) and (b), respectively. The data recorded in the low-intensity and high-intensity measurements are represented by circles and crosses, respectively. The blue solid line corresponds to the SI calculations. The solid and dashed black lines display the predictions for ${\mathrm{Ar}}^{2+}$ by the semiclassical NSDI and SDI models, respectively. Here the unknown phase offset is chosen such that the values of ${\varphi }_0\left(I\right)$ recorded for ${\mathrm{Ar}}^{+}$ at $I=1.5\times {10}^{14}\text{W}{\text{cm}}^{-2}$ agree with the result of the SI calculation. Here the calculations are performed using the measured intensity values and a laser pulse duration of 4.5 fs.

Figure 8

The ${\mathrm{Ar}}^{2+}$ to ${\mathrm{Ar}}^{+}$ yield ratio measured in the low- and high-intensity measurements (light blue and dark blue symbols, respectively) is shown together with the calculated one, assuming NSDI (solid line) and SDI (dashed line). The sum of the contribution of both mechanisms is represented by the dotted line. A scaled intensity $I_{\mathrm{th}}=1.55I$ and an excitation probability of $1\%$ were used for the calculations. The calculated double-ionization yields were averaged over the focal volume assuming a Gaussian intensity profile for the two limiting cases of a 3D integration (red lines) and a 2D integration (blue lines) over the focal volume.