Resolving multiple rescatterings in high-order-harmonic generation

We explore the conditions for resolving high-order electronic recollisions in high-order-harmonic spectroscopy. We identify intrinsic phase mismatch and time-frequency uncertainty as the two fundamental limitations against the spectral distinguishability of these multiple rescatterings. Our numerical computations show that flat-top mid-infrared driving fields are the optimal candidates for the study of multiple recollision phenomena.

Figure 1

(a) Sample of electronic trajectories in a 6-cycle (full width) laser field with a ${cos}^2$-envelope. The grey-dashed line represents the electric field in arbitrary units and the green line indicates the core position. Four pairs of trajectories are represented for energies at first-order recollision of $2.9U_p$ (purple), $2.5U_p$ (dark pink), $1.5U_p$ (pink), and $U_p$ (light pink), whereas the most energetic trajectory, raising $3.17U_p$ at the first recollision, is represented in blue. The vertical axis represents the distance from the core in atomic units ($a_0$). (b) Time-frequency analysis for the HHG spectrum obtained solving TDSE in helium (color background). The laser pulse is modeled as in panel (a) with a wavelength of 2 $\mu m$ and peak intensity of $5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. We use a Gaussian window with spectral FWHM of $\mathrm{\Delta }\omega =10{\omega }_0$, (where ${\omega }_0$ is the fundamental frequency), which corresponds to a temporal FWHM of $\mathrm{\Delta }t=T/11.4$, where $T$ is the laser period. The photon energy of the harmonics is given in terms of the rescattering energy (in units of $U_p$, left vertical axis), and in terms of the harmonic order (right vertical axis). The classical returning kinetic energies at the instant of recollision are shown in black dots for the electrons ionized in the temporal interval between -0.75T and -0.5T. The labels in different colors represent the families of rescattering events that correspond to the wave-packet that was ionized during the same half-cycle.

Figure 2

(a) Transversal and (b) longitudinal spatial field (solid lines) and phase (dashed lines) profiles at $\lambda =2.0\mu \mathrm{m}$, for a BB with $a=95\mu \mathrm{m}$ (green), a Gaussian beam ($M=1$, blue), and a FBM ($M=20$, red), using a beam waist of $W_{0,1}=63.2\mu \mathrm{m}$ that gives a Rayleigh distance of $z_{0,1}=6.28$ mm for the later two cases. (c) Time-frequency analysis of the HHG radiation from the single-atom SFA+ calculation for a laser pulse of $2.0\mu \mathrm{m}$, using $\mathrm{\Delta }\omega =10{\omega }_0$. The rest of parameters of the laser pulse are as in Fig. 1. In panels (d) to (f) we show the results including macroscopic 3D propagation when using a Gaussian beam (d), a BB (e), and a FTB (f). The 3D propagated harmonics are generated in a helium gas cell of thickness ($L$) of 0.1 mm (Gaussian beam), 2 mm (BB), and 1 mm (FTB), at a pressure of 5 torr (gas density of $1.7\times {10}^{17}\mathrm{atoms}/{\mathrm{cm}}^3$). In the Gaussian beam case (d), the gas cell is placed 3 mm after the focus to achieve favorable phase-matching conditions, and the time-frequency yield is multiplied by a factor of 20 for better comparison with panels (e) and (f). In the case of the FTB (f), the gas cell is placed at the focus.

Figure 3

HHG spectra (first row) and time-frequency analysis (second to fourth rows) driven in helium at $0.8\mu \mathrm{m}$ (first column) and $2\mu \mathrm{m}$ (second column). The laser pulse is modeled by a ${cos}^2$ envelope, 6 cycles full width, and peak intensity of $5\times {10}^{14}\text{W}/{\mathrm{cm}}^2$. From row to row, the spectral ($\mathrm{\Delta }\omega$) and corresponding temporal ($\mathrm{\Delta }t$) FWHM of the Gaussian windows in the TFA are ($3.5{\omega }_0, T/4$), ($7{\omega }_0, T/8$), and ($14{\omega }_0, T/16$), with ${\omega }_0$ being the fundamental frequency and $T$ being the laser period. Note that both the spectra and TFA are in logarithmic scale.

Figure 4

(a) Single-atom HHG attosecond pulse train obtained from the Fourier transform of the single-atom harmonic spectrum presented in Fig. 3. Panels (b) and (c) show the propagated spectra (in logarithmic scale) of two individual harmonic bursts (b, $0.75<t_0/T<1.25$ and c, $1.25<t_0/T<1.75$) for the Bessel (green) and flat-top (green) beams considered before at $2\mu \mathrm{m}$. The rest of parameters are the same as in Figs. 2 and 2.