Ellipticity effects and the contributions of long orbits in nonsequential double ionization of atoms

Using a semiclassical model based on tunneling, electron kinematics in the laser field, and phase space volume, we evaluate the ion and electron momentum distributions in nonsequential double ionization of a model atom by an elliptically polarized laser field. For ellipticities exceeding approximately 0.3, we find that the shortest quantum orbit (having the shortest travel time) no longer dominates the double-ionization rate, in contrast to the case of linear polarization. Simultaneously we observe significant violations of symmetry patterns in the ion-momentum distributions and the electron-electron momentum correlations, compared with the case of linear polarization. These violations are very sensitive to the laser pulse duration because of the absence of the late returns for very short pulses. Some of these qualitative effects in the photoelectron distributions should be experimentally detectable. Observation would verify the significance of the late returns and the underlying quantum-orbit concept.

Figure 1

(a) Dimensionless return time $\omega t_n^{\prime }/\pi$, (b) initial transverse velocity scaled by the major axis of the polarization ellipse, $\sqrt{1+{\xi }^2}{v}_0/P_F$ with $P_F=F_0/\omega$, (c) the product of the tunneling rate and the spreading factor ${\left(t_n^{\prime }-t\right)}^{-3}$ in arbitrary units, and (d) the electron kinetic energy upon its return in units of the ponderomotive energy $E\left(t\right)/U_P$ are shown as functions of the dimensionless ionization time $\omega t/\pi$. Results are only shown for $0.5\pi \le \omega t\le 0.7\pi$. For $0.5\le \omega t/\pi \lesssim 0.575$ multiple returns exist. The plots include results for the first five returns $n=1,\dots ,5$ as specified in the inset of panel (a). The ellipticity is $\xi =0.3$. In panel (d) the horizontal line shows the scaled second ionization potential $I_2/U_P$ for NSDI of neon $\left(I_2=1.51\text{a}.\text{u}.\right)$ by a Ti:sapphire laser field $\left(\omega =0.057 \text{a}.\text{u}.\right)$ with intensity $8.0\times {10}^{14} \text{W}/{\text{cm}}^2$. Its intersections with the curves $E\left(t\right)/U_P$ determine the intervals $t_{n<}\le t\le t_{n>}$ of the ionization time for which impact ionization of the second electron is classically allowed. The values $t_{2<}$ and $t_{2>}$ are shown for the second return. We refer to the solutions denoted by no. 1 as the first return or the shortest orbit, to all others as “long orbits” [36].

Figure 2

(Color online) Three returning trajectories for the ellipticity $\xi =0.3$, which all start at $\omega t=1.80$ at the origin but with different initial velocities $v_0$, which are reflected in the different gradients at the start time $\omega t=1.80$. When orbit no. 1 returns to the origin, this is the first time it intersects the line $x=0$. For orbit no. 2, it is the second time, for orbit no. 3 the third, and so on. For linear polarization, the initial velocity $v_0$ is zero, and all orbits are identical.

Figure 3

(Color online) Total yield of doubly charged ions calculated from Eqs. (20, 21) for neon as a function of intensity at different ellipticities.

Figure 4

(Color online) The 2D ion-momentum distribution for ionization of neon by the field of a Ti:sapphire laser with intensity $8\times {10}^{14}\text{W}/{\text{cm}}^2$ for different ellipticities. In the upper two rows, the contributions from the first six returns were summed while the third row presents the contribution from the first return only. The numbers in the lower right corner of each panel specify the relative height of the maxima with respect to linear polarization. In each panel, a thin solid line presents a parametric plot of the vector potential for the respective ellipticity. Additional thin solid lines exhibit the boundaries of the classically allowed regions for those returns whose numbers $n$ are given next to the respective curves.

Figure 5

Ion-momentum distributions along the major axis [(a) and (c)] and the minor axis [(b) and (d)] of the laser polarization ellipse for different ellipticities [explained in the inset of (b)] for the parameters of Fig. 4. The contributions of the first six returns are accounted for in (a) and (b), while (c) and (d) show the corresponding distributions with only the first return included. The curves are normalized to unity at their maxima. The momenta are scaled by $P_F=F_0/\omega$.

Figure 6

(Color online) The distribution $w\left(\phantom{|}{p}_{1x},p_{2x}|p_{1y}>0,p_{2y}>0\right)$ for various ellipticities evaluated with account of six returns (two upper rows). The third row presents the contribution from the first return only. The numbers in the lower right corners give the relative suppression of the larger maximum in the panel with respect to linear polarization. The boundaries of the classically allowed regions for several specified returns are exhibited and identified by the numbers $n$ next to the respective curve. The insets in the upper left corners show the cuts through the corresponding 2D distributions along the $p_{1x}=p_{2x}$ diagonal. The parameters are those of Fig. 4.

Figure 7

(Color online) The same as Fig. 6 except that the intensity is $4.0\times {10}^{14}\text{W}/{\text{cm}}^2$ and the contributions from the first six returns are accounted for in all panels.

Figure 8

Asymmetry (28) as a function of ellipticity at intensities $8.0\times {10}^{14}\text{W}/{\text{cm}}^2$ (dotted line) and $4.0\times {10}^{14}\text{W}/{\text{cm}}^2$ (solid line). At the higher intensity, a local maximum and minimum are at $\xi =0.18$ and $\xi =0.27$, respectively. At the lower intensity, the maximum and minimum are at $\xi =0.22$ and $\xi =0.42$, respectively, and the asymmetry vanishes at $\xi =0.37$ and $\xi =0.45$.