High-order above-threshold ionization by a few-cycle laser pulse in the presence of a terahertz pulse

In our recent letter [D. B. Milošević, Opt. Lett. 47, 1669 (2022)] we proposed to apply a terahertz pulse to modify the ionization process induced by a strong midinfrared laser pulse. If the vector potentials of the laser and THz fields are comparable, the yield and the maximum energy of the photoelectrons emitted in the strong-field ionization process can be significantly increased. The time delay $\mathrm{\Delta }t$ between the THz pulse and the laser pulse is used as a control parameter. In the present work we further analyze this process, with particular emphasis on high-order above-threshold ionization (HATI). Using our improved strong-field approximation we calculate the ionization amplitude by numerical integration over the electron's ionization and travel times. We apply the quantum-orbit theory and the modified saddle-point method to provide a physical explanation. In addition to the previously considered ionization of Cs atoms by a 3100-nm midinfrared laser, we now show that qualitatively similar results can be obtained for HATI of Ar atoms by a much stronger 800-nm laser pulse. Furthermore, for HATI of Ar by a 3100-nm laser, assisted by a THz pulse of a few-MV/cm strength, we show that the photoelectron energy can be in the keV region.

Figure 1

Differential ionization probability $W_{\mathbf{p}}^{\left(1\right)}$ for the electron emission angle $\theta =0$ as a function of the photoelectron kinetic energy (in units of the laser-field ponderomotive energy) for ionization of Cs atoms by the combined linearly polarized sine-square six-cycle laser and two-cycle THz pulse with the zero carrier-envelope phases. Laser intensity and wavelength are $I_{\mathrm{L}}=3\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and 3100 nm, respectively, while the wavelength of the THz field is 100 times longer and $E_{0\mathrm{T}}=400\mathrm{kV}/\mathrm{cm}$. The results in the absence of the THz field are denoted by “LDR” in the legend. The results in the presence of the THz field are shown for the time delays as denoted in the legend (upper panel: $\mathrm{\Delta }t/T_{\mathrm{L}}=40,60,70,100$; lower panel: $\mathrm{\Delta }t/T_{\mathrm{L}}=110,120,130,150,170$).

Figure 2

Upper panel: Contributions of different pairs of quantum orbits to the differential ionization probabilities for the time delay $\mathrm{\Delta }t=120T_{\mathrm{L}}$, obtained applying the modified saddle-point method to the improved SFA. The contributions presented by dotted lines should be neglected after the respective cutoffs. In the lower part of this panel the ionization probability (in arbitrary units) obtained using all presented saddle-point solutions (red solid line) is compared with that calculated by numerical integration (blue dashed line). Lower panel: Contributions of the quantum orbit which leads to the highest cutoff (quantum orbit “1” in the upper panel), calculated for different values of $\mathrm{\Delta }t/T_{\mathrm{L}}$, as denoted above the corresponding cutoffs and in the legend. The laser and THz pulse parameters are as in Fig. 1.

Figure 3

Saddle-point solutions for the complex times $t_0$ and $t$ in units of the laser-field optical period $T_{\mathrm{L}}$. Solutions correspond to the quantum orbits with the highest cutoff and the time delays $\mathrm{\Delta }t/T_{\mathrm{L}}$ as denoted in the legend. In the left panel the solutions $t_0/T_{\mathrm{L}}$ are shifted down by $0.01T_{\mathrm{L}}$ and $0.02T_{\mathrm{L}}$ for $\mathrm{\Delta }t/T_{\mathrm{L}}=50$ and 40, respectively, while in the right panels the solutions $t/T_{\mathrm{L}}$ are shifted both down and left by $0.01T_{\mathrm{L}}$ and $0.02T_{\mathrm{L}}$ for $\mathrm{\Delta }t/T_{\mathrm{L}}=110$ and 100, respectively. The solutions whose contribution to the ionization probability should be neglected after the cutoff are presented by dashed lines. The laser and THz pulse parameters are as in Fig. 1.

Figure 4

Example of the combined linearly polarized sine-square six-cycle laser and two-cycle THz pulses with the zero carrier-envelope phases. Inset shows the laser electric field. The corresponding vector potential is presented in the main panel by the green solid line. Laser intensity and wavelength are $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and 800 nm, respectively, while the wavelength of the much weaker ($E_{0\mathrm{T}}=0.732\mathrm{MV}/\mathrm{cm}$) THz field is 300 $\mu \mathrm{m}$ (this corresponds to 1 THz). The vector potential of the THz pulse is shown for the time delays as denoted in the legend ($\mathrm{\Delta }t/T_{\mathrm{L}}=150,300,450,600$).

Figure 5

Differential ionization probability for direct ($W_{\mathbf{p}}^{\left(0\right)}$) and rescattered ($W_{\mathbf{p}}^{\left(1\mathrm{F}\right)}$) electrons for $\theta =0$ as a function of the photoelectron kinetic energy for ionization of Ar atoms by the combined laser and THz pulses with the parameters of Fig. 4 and the time delays denoted in the legend. The results in the presence of the laser field alone are denoted by “LD” (“LR”) for the direct (rescattered) electrons.

Figure 6

Differential ionization probability for rescattered electrons $W_{\mathbf{p}}^{\left(1\mathrm{F}\right)}$ for $\theta =0$ as a function of the photoelectron kinetic energy for ionization of Ar atoms by the laser pulse alone (black line denoted by LR) and by the combined laser and THz pulses with the time delay $\mathrm{\Delta }t=120T_{\mathrm{L}}$. Laser intensity is $I_{\mathrm{L}}=1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and the wavelength is 3100 nm. Terahertz field has the frequency 1 THz and the strength 2.838 MV/cm (red line), 5 MV/cm (blue line), and 10 MV/cm (green line).

Figure 7

Maximum photoelectron kinetic energy (in units $U_{\mathrm{pL}}$) as a function of the time delay $\mathrm{\Delta }t$ (in units $T_{\mathrm{L}}$), obtained using classical model from [55]. Laser and THz pulse parameters are as in Fig. 6. The THz field strength in MV/cm is denoted above each curve.