Controlling electron-ion rescattering in two-color circularly polarized femtosecond laser fields

High-harmonic generation driven by two-color counter-rotating circularly polarized laser fields was recently demonstrated experimentally as a breakthrough source of bright, coherent, circularly polarized beams in the extreme ultraviolet and soft-x-ray regions. However, the conditions for optimizing the single-atom yield are significantly more complex than for linearly polarized driving lasers and are not fully understood. Here we present a comprehensive study of strong-field ionization—the complementary process to high-harmonic generation—driven by two-color circularly polarized fields. We uncover the conditions that lead to enhanced electron-ion rescattering, which should correspond to the highest single-atom harmonic flux. Using a velocity map imaging photoelectron spectrometer and tomographic reconstruction techniques, we record three-dimensional photoelectron distributions resulting from the strong-field ionization of argon atoms across a broad range of driving laser intensity ratios. In combination with analytical predictions and advanced numerical simulations, we show that “hard” electron-ion rescattering is optimized when the second-harmonic field has an intensity approximately four times higher than that of the fundamental driving field. We also investigate electron-ion rescattering with co-rotating fields, and find that rescattering is significantly suppressed when compared with counter-rotating fields.

Figure 1

(a) The experimental setup used to study strong-field ionization in two-color $(\omega ,2\omega )$ circularly polarized fields. The experimental apparatus consists of a femtosecond laser system, a Mach-Zehnder interferometer, and a velocity map imaging spectrometer. (b) Experimentally measured 3D photoelectron momentum distributions at different intensity ratios for both co- and counter-rotating fields.

Figure 2

Photoelectron distributions for co-rotating (a–e) and counter-rotating (f–j) fields for various intensity ratios of the 395-nm $\left(2\omega \right)$ and 790-nm $\left(\omega \right)$ driving lasers. The shape of the distributions from co-rotating fields is not highly dependent on the intensity ratio, and the electrons are driven to lower final momenta as the $I_{2\omega }/I_{\omega }$ ratio is increased. For the counter-rotating cases, the shape of the momentum distributions is much more sensitive to the exact intensity ratio. The white, yellow, and red rings correspond to 2, 5, and $10U_P$, respectively.

Figure 3

(a) Photoelectron kinetic energy spectrum resulting from SFI using one-color circularly polarized fields. The peak at $U_{\mathrm{P}}$ was used to calibrate the experimental laser intensities. (b–f) Comparison of the photoelectron spectra for counter- and co-rotating fields for various intensity ratios of the 395- and 790-nm fields. For co-rotating fields, little or no high-energy electron yield is observed. For the counter-rotating cases, an enhancement of high-energy electrons (green shaded region) is observed for $I_{2\omega }/I_{\omega }$ ratios of ∼2, 9, and 18. The kinetic energy of the electron is expressed in units of the ponderomotive energy of the two-color field.

Figure 4

Electric fields (green dashed line) and final electron drift momenta (blue solid line). The dots represent time zero for each field and demonstrate that for counter-rotating fields, a maximum of the electric field corresponds to a minimum in the drift momentum. Conversely, for co-rotating fields, a maximum in the electric field corresponds to a maximum of the final drift momentum. The total intensity of each ratio is $2\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, and the light-blue circle indicates a final drift momentum of $2U_P$. The electric field is scaled in order to be displayed on the same axes as the final drift momentum.

Figure 5

Cutoff energies of the direct electrons for different intensity ratios. The largest separation of the minimum and maximum energies occurs when the $I_{2\omega }/I_{\omega }$ ratio is 4 (circles). These curves apply to both co- and counter-rotating fields. However, photoelectron distributions for counter-rotating fields have the greatest yield at the minimum electron energy (green), while photoelectron distributions for co-rotating fields have the greatest yield at the maximum electron energy (magenta).

Figure 6

Photoelectron distributions of the direct electrons calculated using the two-step classical trajectory (TSCT) plotted on a log scale. The distributions are obtained by weighting the classically derived final drift momenta by the tunnel-ionization rates. The effect of the Coulomb potential is ignored and a Gaussian distribution of initial momenta is assumed. The total intensity for each ratio is $2\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, and the white, yellow, and red rings correspond to 2, 5, and $10U_P$, respectively.

Figure 7

The fraction of electrons that pass within 0.05 nm of the parent ion provides an estimate of the probability for hard electron-ion rescattering and HHG. In the TSCT, this type of electron-ion rescattering is optimized at a $I_{2\omega }/I_{\omega }$ ratio of 4 for counter-rotating fields (green), while for co-rotating fields (magenta), no electrons are driven back in close proximity of the parent ion.

Figure 8

Photoelectron distributions obtained from the direct and rescattering terms of the ISFA simulations. The total (direct+rescattering, solid lines) term of the ISFA agrees well with the experimental data (Fig. 3), showing an enhancement of high-energy electrons (dark green region) that result from hard electron-ion backscattering and are seen in the counter-rotating case when the $I_{2\omega }/I_{\omega }$ ratio is 1, 4, and 8 with a maximum at 4. The rescattering term (dashed and dotted lines) of the ISFA shows an even greater enhancement (light and dark green shaded regions) of the hard rescattered electrons. This effect is masked in the experimental spectra as co-rotating cases can drive the more populous direct and soft rescattered electron to fairly high energies. The total intensity for each ratio is $2\times {10}^{14}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}$, and the kinetic energy of the electron is expressed in units of the ponderomotive energy of the two-color field.

Figure 9

Photoelectron distributions obtained from CTMC simulations. The total photoelectron yield (solid lines) agrees well with the experimental data (Fig. 3), showing an enhancement of high-energy electrons (dark green region) that result from hard electron-ion backscattering and are seen in the counter-rotating case when the $I_{2\omega }/I_{\omega }$ ratio is 1, 4, and 8 with a maximum at 4. The photoelectron yield composed of only hard rescattered electrons (dashed and dotted lines), where an electron is considered rescattered if it passes within 0.05 nm of the parent ion, shows an even greater enhancement (light and dark green shaded regions) of the hard rescattered electrons, agreeing with the ISFA simulations (Fig. 8). The total intensity for each ratio is $2\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, and the kinetic energy of the electron is expressed in units of the ponderomotive energy of the two-color field.

Figure 10

Photoelectron distributions from CTMC simulations for counter-rotating fields, for when the Coulomb potential of the parent ion is not included (a–d) and included (e–h), plotted on a log scale. High-energy rescattered electrons are prominent when the $I_{2\omega }/I_{\omega }$ ratio is 4 and 8 (f,g). Additionally, there is a noticeable rotation of the distributions when the Coulomb potential is included, demonstrating the effect of the parent ion on the electron trajectories. The total intensity for each ratio is $2\times {10}^{14}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}$, and the white, yellow, and red rings correspond to 2, 5, and $10U_P$, respectively.

Figure 11

Photoelectron distributions from CTMC simulations for co-rotating fields, for when the Coulomb potential of the parent ion is not included (a–d) and included (e–h), plotted on a log scale. In comparison to the counter-rotating case (Fig. 10), there is still a rotation of the distributions due to the effect of the Coulomb potential on the electron trajectories; however, there is no significant enhancement of high-energy electrons for any intensity ratio. The total intensity for each ratio is $2\times {10}^{14}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}$, and the white, yellow, and red rings correspond to 2, 5, and $10U_P$, respectively.

Figure 12

2D photoelectron distributions numerically calculated by solving the 3D time-dependent Schrödinger equation (TDSE). The numerical results are in good agreement with the experimentally measured distributions (Fig. 2). The total intensity for each ratio is $2\times {10}^{14}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}$.

Figure 13

2D photoelectron momentum distributions numerically simulated using the rescattering term of the ISFA simulations for different intensity ratios plotted on a log scale. The ISFA shows significant high-energy rescattering for counter-rotating fields when the $I_{2\omega }/I_{\omega }$ ratio is 4 and 8 (b and c, respectively). The “hole” in the center of the distributions is a result of the energy conserving condition (Eq. (12) of Ref. [54]) that must be satisfied in the ISFA simulations. The total intensity for each ratio is $2\times {10}^{14}\mathrm{W}\mathrm{c}{\mathrm{m}}^{-2}$, and the white, yellow, and red rings correspond to 2, 5, and $10U_P$, respectively.

Figure 14

The fraction of electrons that pass within 0.05 nm on the parent ion, plotted for two-color counter-rotating fields at various driving laser wavelengths. Hard electron-ion rescattering is optimized when the $U_P$ of the fields are the same. In each case, the vertical lines indicate the intensity ratio that provides equal $U_{\mathrm{P}}$ for the two fields.