Generation of multiple attosecond sub-bursts in femtosecond pulse trains by long-wavelength driving lasers

High-order harmonics generated by a long intense femtosecond laser are known experimentally to create attosecond pulse trains (APTs). In the time domain, an APT consists of a sequence of sharp attosecond bursts that are equally separated by each half optical cycle. Here we show that such well-known features can be modified when a longer wavelength driving laser is used. From our simulations, we show that multiple shorter attosecond sub-bursts exist in the femtosecond pulse train within each half optical cycle and the duration of each sub-burst scales approximately as ${\lambda }_0^{-2}$ with the driving laser wavelength ${\lambda }_0$. We show that such sub-bursts can be found using quantitative rescattering model for harmonics generated from a single atom, and their origin is due to the interference of the quantum orbits from first two returns of the recombining electron. We further show that such sub-bursts can be phase matched under proper laser focusing condition and the position of the gas cell, thus, such new features should be observable experimentally.

Figure 1

Time-frequency analysis of single-atom harmonic emission due to first-return orbits (a), the second-return orbits (b), and the orbits of both returns (c). The results for including all orbits are in (d). Note that the values in (a)–(d) are normalized individually. (e) Temporal profiles of the attosecond pulses synthesized from single-atom high harmonics from different orbits in (a)–(d). The width of the attosecond sub-burst is 30 as due to the interference of quantum orbits from the first and second returns. Here the 1600-nm laser has the intensity of $1.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and the total pulse duration is 30 cycles. The QRS model is applied, in which the microscopic wave packet is computed by truncating the electron excursion time in the standard SFA model to select contributions from a specific return.

Figure 2

Time-frequency analysis of single atom HHG driven by lasers with wavelengths of 2000 nm (a), 2400 nm (b), and 3200 nm (c). The intensity and number of cycles of the total duration are kept the same as those in Fig. 1. (d)–(f) Temporal profiles of attosecond pulses by spectral filtering high harmonics from different quantum orbits at different wavelengths as indicated. The widths of attosecond sub-bursts due to the interference of the first and second return orbits are 20, 14, and 8 as at the three different driving wavelengths, respectively. (g) The intensity ratio of HHG due to the second return and first return orbits as a function of the wavelength.

Figure 3

Time-frequency analysis of single-atom HHG obtained from the QRS model (a) and from the TDSE method (b). (c) Temporal profiles of attosecond sub-bursts by spectral filtering high harmonics in (a) and (b). The laser parameters are the same as those in Fig. 1 except that the FWHM duration is 3 optical cycles.

Figure 4

(a) Induced dipole phase as a function of laser intensity for different quantum orbits (indicated by S1, L1, ...) calculated using classical trajectories. The distributions of quantum orbits characterized by the laser intensity $I_0$ and the phase coefficient $\alpha$ simulated by the SFA theory (b) and by the TDSE method (c). Note that the QRS model gives the same distribution of quantum orbits as the SFA model at a given harmonic order. The wavelength of the driving laser is 800 nm and the harmonic order is H23.

Figure 5

Distributions of quantum orbits simulated by the SFA model at the laser wavelengths of 1200 nm (a), 1600 nm (b), and 2000 nm (c) for the selected harmonic orders. The corresponding induced dipole phases calculated by the classical trajectory for different quantum orbits are shown in (d)–(f).

Figure 6

Map of coherence length of harmonic H65 generated by different quantum orbits. From top to bottom rows, the first, second, and third returns are plotted. The left column is for “short” orbits, and the right column is for “long” ones. The red arrow indicates the direction of the harmonic wave vector. The peak intensity and the beam waist at the focus of the 1600-nm Gaussian beam are $1.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ and $130\mu \mathrm{m}$, respectively.

Figure 7

Same as Fig. 6 but for a 2000-nm driving laser with a beam waist of $200\mu \mathrm{m}$ and a harmonic order of 131.

Figure 8

(a) Intensity distribution of the macroscopic HHG in the far field generated by including all quantum orbits. Intensity distributions of the far-field HHG generated by individual “short” orbits from the first three returns are shown in (b)–(d), and those results by individual “long” orbits are plotted in (f)–(h). (e) HHG spectra by integrating the harmonic yield over the divergence angle are shown for quantum orbits from different returns. The driving laser has a wavelength of 1600 nm; other laser and macroscopic parameters can be seen in the text.

Figure 9

Similar to Fig. 1 but for the macroscopic HHG at 1 mrad in the far field.

Figure 10

(a) Intensity profile of the far-field HHG obtained by including all quantum orbits. (b) Time-frequency analysis of the corresponding far-field HHG at 1 mrad. (c) Temporal profiles of attosecond sub-bursts with the duration of 22 as obtained by spectral filtering far-field harmonics at 1 mrad. The driving laser field has the wavelength of 2000 nm with the peak intensity of $1.5\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. See text for other simulation parameters.