Universality of Returning Electron Wave Packet in High-Order Harmonic Generation with Midinfrared Laser Pulses

We show that a returning electron wave packet in high-order harmonic generation (HHG) with midinfrared laser pulses converges to a universal limit for a laser wavelength above about $3\mu \mathrm{m}$. The results are consistent among the different methods: a numerical solution of the time-dependent Schrödinger equation, the strong-field approximation, and the quantum orbits theory. We further analyze how the contribution from different electron “trajectories” survives the macroscopic propagation in the medium. Our result thus provides a new framework for investigating the wavelength scaling law for the HHG yields.

Figure 1

(a) Typical flattop electric field used in our calculation. (b) Typical time-frequency trace of the induced dipole acceleration from TDSE. (c) Typical returning electron wave packet and its Bezier curve. (d) TDSE results for electron wave packet $|W(\tilde{E}){|}^2$ due to first return at fixed scaled energies of 2.5, 2.0, and 1.5 vs laser wavelength from 0.8 to $3.2\mu \mathrm{m}$. QO results are also shown (symbols), from 0.8 to $6\mu \mathrm{m}$. All results here are for atomic hydrogen with laser intensity of $1\times 10^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 2

(a) TDSE results for rescaled wave packets $|\tilde{W}(\tilde{E}){|}^2$ from atomic hydrogen due to the first return with different laser wavelengths. The curves have been smoothed out using Bezier interpolation. (b) Same as (a), but for short trajectories. (c) and (d) Same as (a) and (b), respectively, but from QO theory.

Figure 3

(a) Wave packets due to the first return from the TDSE for different atoms as shown in the labels. See text for laser parameters. The same are shown in the inset, but for the total wave packets. (b) Total rescaled wave packets from the SFA for different targets. Laser intensities and wavelengths are given in the labels. In both (a) and (b) the wave packets from different targets have been normalized to account for the differences in ionization rates.

Figure 4

Macroscopic HHG yields after propagating in the Ar gas jet of 1 mm thick placed at $z=1\mathrm{mm}$ (a) and $z=3.5\mathrm{mm}$ (b) after the laser focus. Single atom induced dipole is calculated by the QO theory, for a sine-squared envelope laser pulse with $1.6\mu \mathrm{m}$ wavelength, $1\times 10^{14}\mathrm{W}/{\mathrm{cm}}^2$ peak intensity (at the center of gas jet), and a 30-cycle total duration. The beam waist is $36\mu \mathrm{m}$.