Off-axis low-energy structures in above-threshold ionization

The velocity map of the above-threshold ionization electron spectrum at long laser wavelength exhibits a characteristic structure normal to the laser polarization, which has the appearance of a trident or a three-pronged fork. The forklike structure vanishes for few-cycle laser pulses. It is explained in terms of the classical-electron-trajectories model of strong-field ionization augmented so as to allow for rescattering. The analysis reveals its relation to the so-called low-energy structure, which was recently observed for very small transverse momenta.

Figure 1

The logarithm of the photoelectron momentum distribution of the ionization of Xe by a 75-fs (12.5 cycles) laser pulse with $1.8\text{−}\mu \mathrm{m}$ center wavelength. The structure of interest is around $p_y\sim \pm 0.5$ a.u. The labels with arrows mark the characteristic features that are observed at long wavelength, i.e., the LES, the spiderlike structure, and the fork. Panels (c) and (d) show the same as (a) and (b) but for an 11-fs (2 cycles) laser pulse. In (b) and (d), the color scale is clipped from the side with the high count rate for better visibility of the fork. The carrier-envelope phase is not fixed and the peak intensity was around $8\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2$ for both.

Figure 2

The return energy $E_{\mathrm{ret}}$ [thin solid (black) line, scale on the left], the back-scattering energy $E_{\mathrm{bs}}$ [thick solid (blue/green) line, scale on the left], and the forward scattering energy $E_{\mathrm{fs}}$ [dotted (blue) line, logarithmic scale on the right] as given by Eq. (1) in units of $U_{\mathrm{p}}$ as functions of the travel time $\tau$. The various maxima are labeled as defined in the text. The horizontal dashed (red) line at $8U_{\mathrm{p}}$ marks the border between the energies $E_{\mathrm{bs},n}^{+}$ and $E_{\mathrm{bs},n}^{-}$. The orange dots emphasize the two LES energies (also marked in Fig. 3). The extremal energies $E_{\mathrm{fs},n}^{-}$ of the second (“$-$”) family are identically 0 as the respective trajectories start and return at the extrema of the field with ${\tau }_{\mathrm{fs},n}^{-}=n\pi (n=1,2,\cdots )$. Note [by calculation from Eq. (1) or by inspection] that at the times $\tau \equiv {\tau }_{\mathrm{fs},n}^{+}$ we have $E_{\mathrm{ret}}=0,E_{\mathrm{bs}}=E_{\mathrm{fs}}$, and $E_{\mathrm{fs}}$ assumes a maximum, which is given by $E_{\mathrm{fs},n}^{+}$.

Figure 3

Angle-dependent cutoffs of the electron drift energy $E_{\mathrm{kin}}\left(\theta \right)=p^2\left(\theta \right)/2$, Eq. (2). On the $p_x$ axis, we see the cutoffs corresponding to $10.01U_{\mathrm{p}}$ and then to $8.77U_{\mathrm{p}},8.48U_{\mathrm{p}}$, etc. (blue curves, labeled $m^{+}=0,1,2,\cdots$) above $8U_{\mathrm{p}}$, and to $7.03U_{\mathrm{p}},7.45U_{\mathrm{p}}$, etc. (green curves, labeled $m^{-}=1,2,\cdots$) below $8U_{\mathrm{p}}$. The two sets converge to two “limit circles,” one for $p_x>0$ and the other for $p_x<0$, which intersect the vertical axis at $p_x=\pm \sqrt{2\times 8U_{\mathrm{p}}}=2A_0$ and are represented by the dashed (red) curves. They are exact circles and are tangential to the horizontal axis at the origin. Near the origin, for $p_y=0$, we see the LES energies at $0.094U_{\mathrm{p}},0.033U_{\mathrm{p}},\cdots$ marked by solid circles (orange). The two insets show several longitudinal electron trajectories. For details see the text. The orange ellipse at the center of the figure identifies the region relevant for the fork.

Figure 4

The logarithm of the differential ionization yield (in arb. units) of Xe presented in false colors in the electron momentum plane for ionization by a linearly polarized laser pulse having the wavelength $1.8 \mu \mathrm{m}$ and the intensity $5\times {10}^{13}{\mathrm{W}/\mathrm{cm}}^2$. The left panel is calculated for an infinitely extended monochromatic plane wave (CW); the right panel is for a finite pulse of five optical cycles total duration (11 fs FWHM) with a ${cos}^2$ envelope and averaged over the carrier-envelope phase. Focal averaged results are obtained using the improved SFA (with both direct and rescattered electrons). Note that the fork is washed out for the short pulse while it is clearly visible for the cw pulse.