Controlling electron quantum paths for generation of circularly polarized high-order harmonics by ${\mathrm{H}}_2{}^{+}$ subject to tailored ($\omega$, $2\omega$) counter-rotating laser fields

Recently, studies of high-order harmonics (HHG) from atoms driven by bichromatic counter-rotating circularly polarized laser fields as a source of coherent circularly polarized extreme ultraviolet (XUV) and soft-x-ray beams in a tabletop-scale setup have received considerable attention. Here, we demonstrate the ability to control the electron recollisions giving three returns per one cycle of the fundamental frequency $\omega$ by using tailored bichromatic ($\omega$, $2\omega$) counter-rotating circularly polarized laser fields with a molecular target. The full control of the electronic pathway is first analyzed by a classical trajectory analysis and then extended to a detailed quantum study of ${\mathrm{H}}_2{}^{+}$ molecules in bichromatic ($\omega$, $2\omega$) counter-rotating circularly polarized laser fields. The radiation spectrum contains doublets of left- and right-circularly polarized harmonics in the XUV ranges. We study in detail the below-, near-, and above-threshold harmonic regions and describe how excited-state resonances alter the ellipticity and phase of the generated harmonic peaks.

Figure 1

Classical trajectories for $x$, $y$, and $r$ domains versus release time. (a) Scanning the release time $t$ in the interval corresponding to the center of the pulse (22.0–27.3 fs). The gray filled box shows three returns per $T_2$ optical cycle (2.64 fs). (b) The trajectories of the three returns (threefold rosette shape) are shown for the total pulse duration (45 fs) in the $x$ and $y$ domain. The bichromatic frequency components have different peak field strengths corresponding to the intensities of $I_1=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and $I_2=1.25\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 2

Time-dependent electric field of the driving laser pulse. (a) The red dotted and blue solid lines represent the electric field in the $x$ and $y$ direction, respectively. The laser pulse has a duration of 34 optical cycles ($\sim 45$ fs) for the ${\omega }_1$ (395 nm) component and 17 optical cycles ($\sim 45$ fs) for the ${\omega }_2$ (790 nm) component. The bichromatic frequency components have different peak field strengths corresponding to the intensities of $I_1=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and $I_2=1.25\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. (b) The counter-rotating laser fields in the $x\text{−}y$ polarization plane for the total time duration of 45 fs.

Figure 3

HHG spectrum $S\left(\omega \right)$ in the $x$, $y$, and total domain of the ${\mathrm{H}}_2{}^{+}$ molecule in the counter-rotating circularly polarized laser pulses. Circularly polarized HHG spectrum (a) up to $\sim \mathrm{H}81$, (b) below-threshold (H1-H20), (c) above-threshold; plateau region (H20-H53), and (d) above-threshold; plateau and near cutoff region (H55-H74). The laser pulses have a time duration of 34 optical cycles ($\sim 45$ fs) for ${\omega }_1$ (395 nm) and 17 optical cycles ($\sim 45$ fs) for ${\omega }_2$ (790 nm). The green vertical dashed line indicates the corresponding ionization threshold ($I_p$) marked by $1{\sigma }_g$ threshold (H19.13). Resonance A and B in panel (b) correspond to the transitions $1{\sigma }_g-1{\pi }_u$ (H11.7) and $1{\sigma }_g-2{\pi }_u$ (H15.6), respectively. All spectra show a doublet structure, located at positions predicted by energy and spin angular-momentum conservation [filled maroon circles (790 nm) and filled teal squares (395 nm)]. The separation within each doublet is ${\omega }_1-{\omega }_2={\omega }_2$, and different doublets are separated by ${\omega }_1+{\omega }_2=3{\omega }_2$. The bichromatic frequency components have different peak field strengths corresponding to the intensities of $I_1=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and $I_2=1.25\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 4

SST time-frequency analysis $|{\tilde{S}}_{\mathrm{tot}}|$ of the HHG spectra of ${\mathrm{H}}_2{}^{+}$. The horizontal yellow solid line indicates the ionization potential $I_p$ at $\sim \mathrm{H}19$. The vertical white dashed lines indicates one cycle ($T_2$) of the ${\omega }_2$ field (2.64 fs). Three returns per $T_2$ optical cycle are observed, as proposed by the controlled classical electron trajectories in Figs. 1 and 1. The laser parameters used are the same as those in Fig. 3. The color scale is logarithmic.

Figure 5

(a) Ellipticity and (b) phase shift between the $x$ and $y$ components of the harmonic field (below-threshold harmonics) from ${\mathrm{H}}_2{}^{+}$ as a function of harmonic order. The laser parameters used are the same as those in Fig. 3. The filled maroon circles and filled teal squares mark the harmonic peak positions within each doublet. The open blue oval marks the $1{\sigma }_g\text{−}1{\pi }_u$ (H11.7) and $1{\sigma }_g\text{−}2{\pi }_u$ (H15.6) excited-state resonance peaks shown in Fig. 3.

Figure 6

(a) Ellipticity and (b) phase shift between the $x$ and $y$ components of the harmonic field (above-threshold harmonics; plateau region) from ${\mathrm{H}}_2{}^{+}$ as a function of harmonic order. The laser parameters used are the same as those in Fig. 3. The filled maroon circles and filled teal squares mark the harmonic peak positions within each doublet.

Figure 7

(a) Ellipticity and (b) phase shift between the $x$ and $y$ components of the harmonic field (above-threshold harmonics; cutoff region) from ${\mathrm{H}}_2{}^{+}$ as a function of harmonic order. The laser parameters used are the same as those in Fig. 3. The filled maroon circles and filled teal squares mark the harmonic peak positions within each doublet.