Extended quantitative rescattering model for simulating high-order harmonic streaking spectra by synchronization of an intense IR laser and a time-delayed attosecond XUV pulse

We theoretically investigate the modulated high-harmonic generation (HHG) driven by an intense few-cycle infrared (IR) laser field and a weak extreme-ultraviolet (XUV) pulse at a delayed time. We establish an extended quantitative rescattering (EQRS) model to simulate the HHG streaking spectra, with the ideas of correcting the IR ionization and the transition from the ground to continuum states in the strong-field approximation. The EQRS model has an accuracy comparable to that from “exactly” solving the time-dependent Schrödinger equation (TDSE). We reveal that the fringes in the streaking spectra are caused by the interference between the attosecond XUV pulse and harmonics resulting from different recombination pathways under the intense IR laser. We then demonstrate that the XUV pulse can be accurately retrieved by treating the single-atom TDSE results or macroscopic propagation results as the “input” data. This work provides with a tool for efficiently simulating and thoroughly analyzing the XUV-assisted HHG, which could also enhance its capability for tracing the electron dynamics involved in the strong-field phenomena.

Figure 1

(a) The spectra of $|x_2^{\text{SFA}}{(\omega ,I,\tau )|}^2$ calculated at different IR intensities indicated in units of $I_0$, where $I_0={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The peak value is gradually increased with the increase of the IR intensity. (b) Their peak values (black squares) and the fitting curve (red line) as a function of $N\left(I\right)$. (c) The spectra of $|x_2^{\text{SFA}}{(\omega ,I,\tau )|}^2$ at different XUV central photon energies. The peak position coincides with the central energy of the XUV as labeled. (d) Their peak values (black squares) and the fitted curve (red line) with $\left|d\right({\omega }_c\left)\right|$. The insets show the form of power-law fitting functions and the fitted parameters.

Figure 2

The same as Fig. 1, except for calculating $|x_3^{\text{SFA}}{(\omega ,I,\tau )|}^2$.

Figure 3

Comparison of HHG streaking spectra simulated by the EQRS model using hydrogenlike (black solid line) and Gaussian model (red dashed line) potentials for Ne. The spectra are shown at three selected time delays: (a) 0, (b) $-20$, and (c) $-40$ as.

Figure 4

The HHG streaking spectra calculated by four different methods: (a) SFA [black solid line in (e)–(h)], (b) EQRS (red dashed line), (c) TDSE (blue dot-dashed line), and (d) QRS (green dotted line). Comparison of the spectra obtained from the four methods at time delays of (e) 0, (f) 20, (g) 40, and (h) 60 as.

Figure 5

(a) The HHG streaking spectra calculated by the EQRS model, replotted from Fig. 4. The vertical black solid line indicates the cutoff energy $E_c$ of the IR laser, and the slopes of interference fringes are labeled by black dashed, purple dotted, and red dot-dashed lines. The delay-dependent intensity signals of the HHG spectra are shown (b) below and (c) above $E_c$. (d) The electric field of the IR laser (black solid line) and the harmonic emission time as a function of photon energy calculated by the classical trajectory model (red dotted line). $t_{IR}$ is the harmonic emission time, and $\tau$ is the time delay between the XUV pulse and IR laser.

Figure 6

The spectra of $|x_2^{\text{EQRS}}(\omega ,\tau )+x_3^{\text{EQRS}}{(\omega ,\tau )|}^2$ calculated by using the EQRS model with different spectral phases of the XUV pulse defined by the GDD (in atomic units): (a) ${\xi }_1=-0.006$ [black solid line in (e)–(h)], (b) ${\xi }_2=-0.012$ (red dashed line), (c) ${\xi }_3=-0.018$ (blue dot-dashed line), and (d) ${\xi }_4=-0.024$ (green dotted line). Black dotted lines are the contour lines which indicate 20% of the peak spectral intensity. (e) The phase of $x_2^{\text{EQRS}}(\omega ,\tau )+x_3^{\text{EQRS}}(\omega ,\tau )$ at zero time delay. (f) The HHG spectra including the interference with $x_1^{\text{QRS}}\left(\omega \right)$ at zero time delay. The streaking spectra versus the time delay at two fixed photon energies: (g) 62 and (h) 76 eV.

Figure 7

Characterization of three XUV pulses centered at 71.3 eV with a bandwidth of 9 eV. The GDD coefficients $\xi$ (in atomic units) of these XUV pulses are 0 (first row), $-0.016$ (second row), and $-0.024$ (third row). (a)–(c) The input modulated spectra simulated by the TDSE are treated as experimental data. (d)–(f) Comparison of the input and retrieved spectral phases. (g)–(i) Comparison of the input and retrieved XUV intensity profiles in time domain. (The black solid lines are the input data, while the red dashed lines are retrieved results.) The spectral intensity $U{\left(\omega \right)}^2$ of the XUV pulse (blue dotted line) is assumed to be known, as shown in (d).

Figure 8

(a) The HHG streaking spectra after accounting for the propagation effects by averaging IR intensities (input). The GDD coefficient $\xi$ (in atomic units) is $-0.016$. (b) The retrieved modulation spectra using the single-atom theory. Comparison of (c) the retrieved XUV spectral phase and (d) the temporal intensity profile.