Nonlinear Dichroism in Back-to-Back Double Ionization of He by an Intense Elliptically Polarized Few-Cycle Extreme Ultraviolet Pulse

Control of double ionization of He by means of the polarization and carrier-envelope phase (CEP) of an intense, few-cycle extreme ultraviolet (XUV) pulse is demonstrated numerically by solving the six-dimensional two-electron, time-dependent Schrödinger equation for He interacting with an elliptically polarized XUV pulse. Guided by perturbation theory (PT), we predict the existence of a nonlinear dichroic effect ($\propto I^{3/2}$) that is sensitive to the CEP, ellipticity, peak intensity $I$, and temporal duration of the pulse. This dichroic effect (i.e., the difference of the two-electron angular distributions for opposite helicities of the ionizing XUV pulse) originates from interference of first- and second-order PT amplitudes, allowing one to probe and control $S$- and $D$-wave channels of the two-electron continuum. We show that the back-to-back in-plane geometry with unequal energy sharing is an ideal one for observing this dichroic effect that occurs only for an elliptically polarized, few-cycle attosecond pulse.

Figure 1

(a) Sketch of two-electron energy spectra produced when He absorbs one or two photons from a single ten- or three-cycle pulse with $\omega =65\mathrm{eV}$. For the three-cycle pulse, the one- and two-photon perturbation theory amplitudes ($A_1$ and $A_2$) overlap at $\approx 4\mathrm{eV}$ above the DPI threshold at 79 eV owing to the large pulse bandwidth. (b) The in-plane back-to-back (BTB) geometry for DPI of He with electron momenta ${\mathbf{p}}_1$ and ${\mathbf{p}}_2$ in the polarization plane orthogonal to the laser propagation direction $\widehat{\mathbf{k}}$. The major and minor axes of the polarization ellipse are defined by the unit vectors $\widehat{\mathit{\epsilon }}$ and $\widehat{\mathit{\zeta }}$. The BTB angle, $\phi$, is the angle between ${\widehat{\mathbf{p}}}_1$ and $\widehat{\mathit{\epsilon }}$.

Figure 2

The TDP $\mathcal{W}(\widehat{\mathbf{p}},\mathbf{e})$ [Eq. (6)] (in units of ${10}^{-5}\mathrm{a}.\mathrm{u}.$) vs. $\phi$ [cf. Fig. 1] for DPI of He by a three-cycle XUV pulse (with $\omega =65\mathrm{eV}$, $I=2\mathrm{PW}/{\mathrm{cm}}^2$, $T=190.9$ as, a ${\cos }^2$ envelope, and an ellipticity $\eta =\pm 0.5$ or $\xi =\pm 0.8$) for four CEPs: (a) $\varphi =0$, (b) $\varphi =\pi /3$, (c) $\varphi =\pi /2$, (d) $\varphi =5\pi /6$. All results are for the back-to-back geometry and unequal energy sharing (UES): $E_1=0.7\mathrm{eV}$ and $E_2=3.3\mathrm{eV}$. In (c) we give for comparison ${\mathcal{W}}^{(L=1)}(\mathbf{p},\mathbf{e})$; see text for discussion.

Figure 3

Angular dependence of the dichroism, $\mathrm{\Delta }{\mathcal{W}}_{\xi }$, for two CEPs: (a) $\varphi =0$, and (b) $\pi /2$. The contributions of $\mathrm{\Delta }{\mathcal{W}}_{\xi }^{(L=1)}\approx \mathrm{\Delta }{\mathcal{W}}_{D1}$, $\mathrm{\Delta }{\mathcal{W}}_{\xi }^{(L=0,2)}=\mathrm{\Delta }{\mathcal{W}}_{D2}$, and $\mathrm{\Delta }{\mathcal{W}}_{D12}$ are also shown (see text for discussion). (c) Ellipticity dependence of $\mathrm{\Delta }{\mathcal{W}}_{\xi }$. (d) Pulse intensity-dependence of $\mathrm{\Delta }{\mathcal{W}}_{\xi }$; results are scaled by $(I/I_r{)}^{3/2}$, where $I_r=2\mathrm{PW}/{\mathrm{cm}}^2$. Unless otherwise specified, $\varphi =\pi /2$, $\xi =0.8$, $I=2\mathrm{PW}/{\mathrm{cm}}^2$, $E_1=0.7\mathrm{eV}$, and $E_2=3.3\mathrm{eV}$.

Figure 4

Angular dependence of the relative dichroism (RD), $\mathrm{\Delta }{\mathcal{W}}_{\xi }/[\mathcal{W}(\widehat{\mathbf{p}},\mathbf{e})+\mathcal{W}(\widehat{\mathbf{p}},{\mathbf{e}}^{*})]$, for the BTB geometry [cf. Fig. 1]. Unless otherwise specified, $\varphi =\pi /2$, $\xi =0.8$, $I=2\mathrm{PW}/{\mathrm{cm}}^2$, $E_1=0.7\mathrm{eV}$, and $E_2=3.3\mathrm{eV}$. The panels show its sensitivity to (a) the CEP, (b) the energy sharing, (c) the ellipticity, and (d) the pulse intensity.