Conditions for perfect circular polarization of high-order harmonics driven by bichromatic counter-rotating laser fields

Recently, studies of high-order harmonic generation (HHG) from atoms driven by bichromatic counter-rotating circularly polarized laser fields have received considerable attention for this process could be a potential source of coherent circularly polarized extreme ultraviolet (XUV) and soft-x-ray beams in a tabletop-scale setup. In this paper, we address the problem with molecular targets and perform a detailed quantum study of the ${\mathrm{H}}_2{}^{+}$ molecule in bichromatic (${\omega }_0, 2{\omega }_0$) counter-rotating circular polarized laser fields where we adopt wavelengths (790 and 395 nm) and intensities ($2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$) reported in a recent experiment [K. M. Dorney et al., Phys. Rev. Lett. 119, 063201 (2017)]. Here, we demonstrate appropriate conditions to produce perfectly circular polarized harmonics. The calculated radiation spectrum contains doublets of left and right circularly polarized harmonics which display perfect circular polarization with use of the trapezoidal pulse shape, and substantial deviations from perfect circular polarization with use of the sine-squared pulse shape. We also study in detail short- and long-cycle counter-rotating circularly polarized driving pulses with a time delay between the two driving fields, ${\omega }_0$ and $2{\omega }_0$. These time delayed circularly polarized driving pulses are applied to H atoms and ${\mathrm{H}}_2{}^{+}$ molecules, and in both atomic and molecular cases we conclude a zero time delay corresponds to the highest HHG intensity for short pulses. For longer pulses there are no distinct differences in HHG intensities between the zero and nonzero time delays if the latter are within a few optical cycles of the fundamental frequency.

Figure 1

Time-dependent (a) sine-squared and (b) trapezoidal electric field of the driving laser pulse (time-delay $\tau =0$). The red dotted and blue dashed lines represent the electric field in the $x$ and $y$ direction, respectively. The laser pulses (a) and (b) have a duration of 17 optical cycles ($\sim 45$ fs) for the ${\omega }_0$ (790 nm) component and 34 optical cycles ($\sim 45$ fs) for the $2{\omega }_0$ (395 nm) component. Both frequency components have the same peak field strength corresponding to the intensity of $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 2

HHG spectrum $S_{\mathrm{tot}}\left(\omega \right)$ as well as contributions $S_x\left(\omega \right)$ and $S_y\left(\omega \right)$ from the $x$ and $y$ projections of the dipole acceleration for the ${\mathrm{H}}_2{}^{+}$ molecule subject to the counter-rotating circularly polarized sine-squared laser pulses. Circularly polarized harmonic doublets (a) up to $\sim \mathrm{H}80$, (b) in the below- and near-threshold region (H1–H26), (c) in the above-threshold plateau region (H27–H53), and (d) in the above-threshold plateau and near cutoff region (H54–H80). The laser pulses have a time duration of 17 optical cycles ($\sim 45$ fs) of frequency ${\omega }_0$ (wavelength 790 nm) and 34 optical cycles ($\sim 45$ fs) of frequency $2{\omega }_0$ (wavelength 395 nm). The black solid, red dotted, and blue dashed lines represent the HHG spectrum in the $S_{\mathrm{tot}}\left(\omega \right), S_x\left(\omega \right)$, and $S_y\left(\omega \right)$ domains, respectively. The green vertical dashed line indicates the corresponding ionization threshold ($I_p$) of the $1{\sigma }_g$ molecular orbital (H19.13). Filled maroon circles and filled teal squares indicate the positions of the peaks with the frequencies $(3n_c+1){\omega }_0$ and $(3n_c+2){\omega }_0$, respectively. The separation between the peaks within each doublet is ${\omega }_0$, and different doublets are separated by $3{\omega }_0$. Both bichromatic frequency components have the same peak field strength corresponding to the intensity of $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 3

Ellipticity of the harmonic radiation from ${\mathrm{H}}_2{}^{+}$ as a function of the harmonic order: (a) below- and near-threshold region (H1–H26), (b) above-threshold plateau region (H27–H53), and (c) above-threshold plateau and near cutoff region (H54–H80). The sine-squared laser pulse parameters used are the same as those in Figs. 1 and 2. The filled maroon circles and filled teal squares mark the peak positions of the harmonics $(3n_c+1){\omega }_0$ and $(3n_c+2){\omega }_0$, respectively, within each doublet.

Figure 4

Phase shift between the $x$ and $y$ components of the harmonic field from ${\mathrm{H}}_2{}^{+}$ as a function of the harmonic order: (a) below- and near-threshold region (H1–H26), (b) above-threshold plateau region (H27–H53), and (c) above-threshold plateau and near cutoff region (H54–H80). The sine-squared laser pulse parameters used are the same as those in Figs. 1 and 2. The filled maroon circles and filled teal squares mark the peak positions of the harmonics $(3n_c+1){\omega }_0$ and $(3n_c+2){\omega }_0$, respectively, within each doublet.

Figure 5

HHG spectrum $S_{\mathrm{tot}}\left(\omega \right)$ as well as contributions $S_x\left(\omega \right)$ and $S_y\left(\omega \right)$ from the $x$ and $y$ projections of the dipole acceleration for the ${\mathrm{H}}_2{}^{+}$ molecule subject to the counter-rotating circularly polarized trapezoidal laser pulses. Circularly polarized harmonic doublets (a) up to $\sim \mathrm{H}80$, (b) in the below- and near-threshold region (H1–H26), (c) in the above-threshold plateau region (H27–H53), and (d) in the above-threshold plateau and near cutoff region (H54–H80). The laser pulses have a time duration of 17 optical cycles ($\sim 45$ fs) of frequency ${\omega }_0$ (wavelength 790 nm) and 34 optical cycles ($\sim 45\mathrm{fs}$) of frequency $2{\omega }_0$ (wavelength 395 nm). The black solid, red dotted, and blue dashed lines represent the HHG spectrum in the $S_{\mathrm{tot}}\left(\omega \right), S_x\left(\omega \right)$, and $S_y\left(\omega \right)$ domains, respectively. The green vertical dashed line indicates the corresponding ionization threshold ($I_p$) of the $1{\sigma }_g$ molecular orbital (H19.13). Filled maroon circles and filled teal squares indicate the positions of the peaks with the frequencies $(3n_c+1){\omega }_0$ and $(3n_c+2){\omega }_0$, respectively. The separation between the peaks within each doublet is ${\omega }_0$, and different doublets are separated by $3{\omega }_0$. Both bichromatic frequency components have the same peak field strength corresponding to the intensity of $2\times {10}^{14}$ $\mathrm{W}/{\mathrm{cm}}^2$.

Figure 6

Ellipticity of the harmonic radiation from ${\mathrm{H}}_2{}^{+}$ as a function of the harmonic order: (a) below- and near-threshold region (H1–H26), (b) above-threshold plateau region (H27–H53), and (c) above-threshold plateau and near cutoff region (H54–H80). The trapezoidal laser pulse parameters used are the same as those in Figs. 1 and 5. The filled maroon circles and filled teal squares mark the peak positions of the harmonics $(3n_c+1){\omega }_0$ and $(3n_c+2){\omega }_0$, respectively, within each doublet.

Figure 7

Phase shift between the $x$ and $y$ components of the harmonic field from ${\mathrm{H}}_2{}^{+}$ as a function of the harmonic order: (a) below- and near-threshold region (H1–H26), (b) above-threshold plateau region (H27–H53), and (c) above-threshold plateau and near cutoff region (H54–H80). The trapezoidal laser pulse parameters used are the same as those in Figs. 1 and 5. The filled maroon circles and filled teal squares mark the peak positions of the harmonics $(3n_c+1){\omega }_0$ and $(3n_c+2){\omega }_0$, respectively, within each doublet.

Figure 8

HHG spectrum $S\left(\omega \right)$ of the hydrogen atom subject to the counter-rotating few-cycle circularly polarized sine-squared laser pulses. The pulse durations measured in optical cycles of the frequency ${\omega }_0$ are $N_1=3$ and $N_2=2$ for the ${\omega }_0$ and $2{\omega }_0$ fields, respectively. Both bichromatic frequency components have the same peak intensity $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The harmonic photon energy range shown is 1 to 5.5 a.u. Solid (black) line: $\tau =0$ (zero time delay); dashed (red) line: $\tau =T_0$ (positive time delay [Eq. (13)] corresponds to the $2{\omega }_0$ field arriving first).

Figure 9

HHG spectrum $S\left(\omega \right)$ of the ${\mathrm{H}}_2{}^{+}$ molecule subject to the counter-rotating few-cycle circularly polarized sine-squared laser pulses. The pulse duration is two optical cycles of the frequency ${\omega }_0$ and peak intensity is $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ for both bichromatic components. Solid (black) line: $\tau =0$ (zero time delay), dotted (red) line: $\tau =1.32\mathrm{fs}$ (positive time delay corresponds to the $2{\omega }_0$ field arriving first), and dashed (blue) line: $\tau =-1.32$ fs (negative time delay corresponds to the ${\omega }_0$ field arriving first). The green vertical dashed line indicates the ionization threshold ($I_p$) of the $1{\sigma }_g$ molecular orbital.