Fig. 1. Laser waveforms (red lines) of (a) five-color field in Ref. [31] and (b) optimized two-color chirped field. In (a) and (b), the excursion distances of the classical electrons are shown (blue dashed lines), leading to the cutoff harmonics (maximum electron kinetic energies indicated), and the green shadow marks the effective waveform. (c)–(e) The waveforms of two-color chirped fields are extended for longer laser durations of 16, 24, and 32 fs. The green regions show the temporal gate for the generation of continuous high harmonics, leading to IAPs. (o.c. means the optical cycle of an 800-nm laser.)

Fig. 2. Comparison of (a) single-atom and (b) macroscopic high-harmonic spectra by using single-color laser (800 nm), five-color field (from Ref. [31]), and two-color chirped field [shown in Fig. 1(c)]. Laser duration is fixed at 16 fs for all driving fields. For the macroscopic calculations, a 1-mm-long gas jet is located 0.5 mm after laser focus, the gas pressure is 10 Torr (distributed uniformly in the jet), and the beam waist is assumed as 50 μm for all colors. For three cases, the single-atom waveforms are only formed in the center of the gas jet, and the harmonic spectra in (b) are thus normalized according to the individual input pulse energy for easy comparison. Time-frequency pictures of harmonic emission and the temporal profiles of attosecond pulses with (c)–(e) two-color chirped and (f)–(h) five-color fields.

Fig. 3. Harmonic spectra, time-frequency analysis of harmonic emission, and temporal profiles of attosecond pulses, similar to Fig. 2. However, the driving lasers are replaced by the two-color chirped fields with durations of 24 and 32 fs [as plotted in Figs. 1(d) and 1(e)]. (c)–(e) 24 fs and (f)–(h) 32 fs. In (e) and (h), the IAPs with durations of 240 and 200 as are obtained by synthesized high-harmonics in the same photon energy region of 130–219 eV.

Fig. 4. (a) Macroscopic harmonic spectra with the two-color chirped field at different gas pressures of 10, 50, and 100 Torr. The pulse duration is 16 fs, and other macroscopic conditions are the same as those in Fig. 2. Time-frequency wavelet analysis of harmonic emission and the generation of IAPs at (b), (c) 50 Torr and (d), (e) 100 Torr. In (c) and (e), 200-as single attosecond pulses are generated by synthesized high-harmonics in slightly different photon-energy regions.

Fig. 5. (a) Optimized two-color chirped laser pulse (using parameters in Table 2) with laser duration of 8 fs when the target cutoff energy is set as 330 eV; (b) extended waveform for the laser duration of 16 fs. The green areas indicate the temporal gate for the generation of soft X-ray harmonics and IAPs. (o.c. is the optical cycle of the 800-nm laser.) (c) Single-atom harmonic spectra by using optimized waveforms with different preset cutoff energies, showing that the cutoff energy is extended with the decrease of harmonic yields; (d) macroscopic harmonic spectrum with the two-color chirped waveforms (800  nm+1600  nm) as shown in (b). In the calculation, a 1-mm-long gas jet is located 3.5 mm after the laser focus, the gas pressure is 10 Torr (distributed uniformly in the jet), and the beam waist is assumed as 40 μm for two colors. (e), (f) Time-frequency analysis of harmonic emission; and (g) time profile of IAP with high harmonics in the photon-energy range of 169–245 eV.