Double photoelectron momentum spectra of helium at infrared wavelength

Double photoelectron momentum spectra of the helium atom are calculated ab initio at extreme ultraviolet and near-infrared wavelengths. At short wavelengths two-photon double-ionization yields, two-electron energy spectra, and triply differential cross sections agree with results from recent literature. At the near-infrared wavelength of $780\mathrm{nm}$ the experimental single-to-double-ionization ratio is reproduced up to intensities of $4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, and two-electron energy spectra and joint angular distributions are presented. The time-dependent surface flux approach is extended to full $3+3$ spatial dimensions and systematic error control is demonstrated. We analyze our differential spectra in terms of an experimentally accessible quantitative measure of correlation.

Figure 1

Spatial partitioning of the wave function: At large times, regions $B,S$ and $\overline{S}$, and $D$ correspond to bound, singly ionized, and doubly ionized parts, respectively. Arrows symbolize the flux across the region boundaries.

Figure 2

Total two-photon DI cross section as a function of the photon energy. Results with $R_c=80\mathrm{a}.\mathrm{u}.$ and pulse duration $4\mathrm{fs}$ and peak intensity $I={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ (solid line) agree with Refs. [9, 28, 29] (dashed lines). Already with $R_c=20\mathrm{a}.\mathrm{u}.$ (dotted line) good qualitative agreement is reached.

Figure 3

(Left) Energy probability distribution $P(E_1,E_2)$ for a $n=120$ cycle pulse with photon energy $\hslash \omega =42\mathrm{eV}$ and peak intensity $I={10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ for a ${cos}^2$ pulse envelope. (Right) Calculation with a ${cos}^8$ envelope of the same FWHM. Structures generated by the ${cos}^2$ sidebands disappear.

Figure 4

Triply differential cross section for two-photon DI of He at photon energy $\hslash \omega =42\mathrm{eV}$ and equal energy sharing for different tSurff radii $R_c$. The same pulse parameters as in Ref. [9] were used.

Figure 5

(Left) Correlated side-by-side emission at three-photon equal energy sharing. (Right) Uncorrelated three-photon sequential ionization at $(E_1,E_2)\approx (1.09,0.64)\mathrm{a}.\mathrm{u}.$ Plots are normalized to ${max}_{E_1,E_2}\left[P(E_1,E_2)\right]=1$.

Figure 6

(Top) Probability distribution $P(E_1,E_2)$ shown in Fig. 3 along the $N=3$-photon line as a function of the energy difference $\mathrm{\Delta }E:=E_1-E_2$. (Bottom) Angular correlation $C$ [Eq. (48)] along the same line. Vertical lines: sequential peaks involving the ionic ground state (solid) and excited ionic states (dashed). The minima for higher ionic excitations are slightly displaced due to the cutoff of the Coulomb potential at $R_c=20\mathrm{a}.\mathrm{u}.$

Figure 7

(Top) Photoelectron spectrum as a function of the shared total energy [Eq. (50)] for two relative time delays between XUV and IR pulses. Large positive time delays correspond to the XUV pulse coming after the IR pulse. The green dashed line indicates $E_{\mathrm{tot}}=60\mathrm{eV}$. (Bottom) $P(E_{\mathrm{tot}}=60\mathrm{eV})$ as a function of the relative time delay: dependence on $R_c$ and comparison with Ref. [36]. Arbitrary units in Ref. [36] are adjusted to approximately match our results.

Figure 8

Ratio of DI yield and single-ionization yield as a function of intensity. Experiment and model [1], model calculations [70], and our full two-electron results for increasing $R_c$. Upward error bars indicate the long pulse limit.

Figure 9

Total DI spectra $P(E_1,E_2)$ [Eq. (44)] for several of the data points with truncation interval $R_c=30\mathrm{a}.\mathrm{u}.$ shown in Fig. 8. Intensities in units of ${10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ are (a) $1.6$, (b) $2.5$, (c) $3.5$, and (d) $4.0$. White dashed lines mark $2U_p$.

Figure 10

Error estimates $\mathcal{E}$ by comparing computations with (a) radii $R_c=25\mathrm{a}.\mathrm{u}.$ vs $R_c=35\mathrm{a}.\mathrm{u}.$ for $I=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and (b) $R_c=25\mathrm{a}.\mathrm{u}.$ vs $R_c=30\mathrm{a}.\mathrm{u}.$ for $I=3.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. Errors are only plotted where $P(E_1,E_2)$ is larger than $1\%$ of its maximum.

Figure 11

Total momentum spectrum $P\left(E_{\mathrm{tot}}\right)$ [Eq. (50)] for a $780\mathrm{nm}n=4$ cycle pulse. (Left) $2.6\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ cutoff at $5.6U_p$. (Right) $3.25\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ cutoff at $6.6U_p$. Solid lines mark the momentum corresponding to $2U_p$; dashed lines mark the transition to the exponential decay at the cutoff of the spectrum; dotted lines indicate the slopes. The behavior conforms with Fig. 3 of [34] for $390\mathrm{nm}$ at the corresponding intensities $1.04\times {10}^{15}\mathrm{W}/{\mathrm{cm}}^2$ and $1.3\times {10}^{15}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 12

JADs for wavelength $780\mathrm{nm}$ and intensity $I=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. Panels (a)–(c) correspond to the energies marked in the $P(E_1,E_2)$ distribution, bottom right panel. Plots are normalized to ${max}_{E_1,E_2}\left[P(E_1,E_2)\right]=1$.

Figure 13

Cuts through JADs of Fig. 12 at ${\theta }_2=0$ for various values of $R_c$. At low energies convergence is only qualitative; see (a) and (c). At high energies the angular distribution stabilizes at $R_c=30\mathrm{a}.\mathrm{u}.$; see (b).

Figure 14

Angular correlation $C$ as a function of photoelectron energies for intensities $1.6\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ (left) and $2.0\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ (right). Computations with $R_c=30$.

Figure 15

Correlated side-by-side emission for DI at $780\mathrm{nm},I=3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, for momenta parallel to the polarization axis. Computation with $R_c=30\mathrm{a}.\mathrm{u}.$ Dashed lines mark $2U_p$, beyond which “fingerlike” structures appear similar as observed in experiment [12].

Figure 16

Peak partial wave populations ${\rho }_{\mathrm{peak}}(l_1,l_2)$ for a pulse with $\lambda =780\text{nm},n=4$, and $I=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 17

(Left) Energy probability distribution $P(E_1,E_2)$ for a $n=120$ cycle pulse with photon energy $\hslash \omega =42\mathrm{eV}$ and peak intensity $I={10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ using a ${cos}^8$ envelope and averaging time $\mathrm{\Delta }T=1\mathrm{ps}$. (Right) Without averaging.