We theoretically investigate the high-order harmonic generation driven by sawtooth-like laser fields. It is found that the coherence of the high-order harmonic spectrum could be controlled by adjusting the sawtooth shape parameter α (0 ≤ α ≤ 0.5). With proper α, the intensity of the high-order harmonic spectra come from the electrons with long trajectories can be greatly weakened and the harmonic photons in the plateau burst with high coherence in a broad frequency range. Selecting these harmonics, a phase-stabilized attosecond pulse train with regular pulse structure could be obtained.
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Till now, high-order harmonic generation(HHG), which comes from the atoms or molecules driven by strong laser fields, has become a famous nonlinear optic phenomenon, due to its widely applications on attosecond pulse generation [1–6] and the detection of ultra-fast processes . In general, the physical origin of HHG can be well described by the following three-step model proposed by Corkum : Firstly the electron is ionized, then accelerated in the laser field and recombines with the parent core. During the recombination, a photon is emitted with energy equal to the return kinetic energy plus the ionization potential. This process occurs in each half cycle of the driver field and there are two dominant quantum paths for each harmonics. Therefore, these harmonics are usually not coherent and the attosecond pulse train(APT) generated by them is not phase-stabilized . However, in the manipulation of ultra-fast processes, phase-stabilized APT is needed . Hence, the coherent control of HHG process in order to obtain an APT with the same carrier-envelop phase (CEP) from pulse to pulse is a topic of great interest. So far, a variety of methods have been proposed for the generation of phase-stabilized APT. Mauritsson et al. found that by adding a second harmonic field an APT with stabilized phase can be obtained . Hong et al.found that under the modulation of a static electric field the quantum path of the electron can be controlled and an APT with stabilized phase can be obtained . Using asymmetric target molecules, the HHG spectrum is phase locked near the cutoff from which a phase-stabilized APT can be obtained, as proposed in Ref. . Besides the stable CEP, the structure of the pulse in the APT need to be considered because the irregular pulse structure of the APT may limit its application. As is known, the irregular pulse structure is due to that the harmonic photons come from different quantum trajectories, i.e., the short and long paths, are not emitted in phase . Thus, in order to generate phase-stabilized APT and regular pulse structure, the effect of the long path should be weakened or eliminated. To realize this, the waveform of the driver field should be well controlled. Due to the development of laser technology, it is possible to generate arbitrary optical waveforms in experiments . Based on this, in this paper we propose a novel method for the generation of coherent HHG in the plateau region to obtain phase-stabilized APT, i.e., using sawtooth-like driver field. The rest of this paper is organized as follows. In Sec. 2 the principle and method are described. Then the numerical results and analysis are presented in Sec. 3 and the conclusions are given in Sec. 4 at last.
2. Principle and method
The interaction between a z-polarized laser field and the helium atom can be described by the following time-dependent Schrödinger equation(TDSE) with the dipole and single active electron approximations,15]. The time dependent wavefunction ψ(r, t) can be obtained by solveing Equation (1) with a Peaceman-Rachford scheme which is described detailed in . Then the HHG power spectrum can be calculated by taking the Fourier transform of the time-dependent acceleration, i.e., 17]
3. Numerical results and analysis
Usually, a sawtooth laser field can be expressed as14]. The amplitudes of the five laser fields can be obtained by Fourier expansion, which can be expressed as Figure 1 shows the electric fields of Es(α, t) and Ef(α, t) for α = 0.25 and 0.5. From this figure, we can see that Ef(α, t) is a good approximation of Es(α, t), i.e., Ef(α, t) keeps the basic shape of Es(α, t). Since Es(α, t) can not be realized in experiment exactly, we only consider the HHG spectrum driven by Ef(α, t).
Figure 2 shows the HHG spectra driven by the laser fields with different α. From this figure, we can see that only odd harmonics can be observed for α = 0.25. For α = 0.35 and 0.5, both odd and even order harmonics appear in the HHG spectra. This is because that the driver field contains even harmonics for α = 0.35 and 0.5 and does not for α = 0.25. In order to investigate how the coherence of the HHG spectrum is related to the sawtooth shape parameter α, we consider attosecond pulses generation from the harmonic spectrum in the plateau region. In Fig. 3, we show the attosecond pulse trains obtained from the HHG spectra in Fig. 2. As can be seen from this figure, for α = 0.25 the pulses appear in each half cycle and each of them has irregular side peaks. For α = 0.35, there are two types of pulses A and B in the pulse train (for concision, only two pulses are marked) and both of them burst in each optical cycle, as shown in Fig. 3(b). It can be clearly seen that the intensities of A type pulses are much higher than those of B type pulses. For α = 0.5, a pulse train is obtained with a time interval of T and pulse duration of 147as, as is displayed in Fig. 3(c). The time profiles of the pulses are similar and no side peaks appear, which means that the selected harmonics are of high coherence.
In order to explore the physics underlying the above phenomena, we perform the time-frequency analysis of the HHG spectra using the wavelet transform of the dipole acceleration  and the results are shown in Fig. 4. As can be seen from Fig. 4(a), the harmonic photons are emitted in each half cycle for α = 0.25. For each peaks, there are two dominant quantum paths for each harmonic, i.e., the short path(the positive-slope section) and long path(the negative-slope section). Due to the interference between the two paths, the harmonics are not emitted in phase which is responsible to the irregular pulse structure of the APT in Fig. 3(a). For α = 0.35, there are two types peaks in each optical cycle which are marked with A and B (for concision, only two peaks are marked, the same below), as is displayed in Fig. 4(b). Apparently, the intensities of A peaks are much higher than those of B pulses. So two types of pulses are obtained by selecting the harmonics between the 20th- and 40th-order. Though the intensities of the long paths of A peaks are significantly lower than those of the short paths, they have apparent effects on the coherence of the HHG spectrum which lead to the side peaks of A type pulses in Fig. 3(b). For α = 0.5, as shown in Fig. 4(c), there are still two types of peaks (A′ and B′) with different cutoffs in each optical cycle.The cutoffs of A′ and B′ peaks are at about the 60th- and 90th-order harmonic, respectively. However, for the harmonics below the 60th order, the intensities of A′ type peaks are much higher than those of B′ type peaks. Moreover, the intensity of the long paths of A′ type peaks are so weaker that they nearly can not be seen. Therefore, the long paths have little contributions to the HHG spectrum which means the corresponding HHG spectrum is of high coherence. So, each pulse in the pulse train of Fig. 3(c) has a regular shape. From the analysis above, we can see that the coherence of the HHG spectrum can be controlled by adjusting the sawtooth shape parameter α.
For a further investigation, we consider the pulse train generated from different harmonic ranges in the plateau since the HHG spectrum is highly coherent for α = 0.5. Figure 5 shows the electric fields obtained by selecting different harmonics ranges. From this figure, we can see that the time evolution of the pulses in each pulse train are the same though the center frequency increases from (a) to (c). This means that the phase-stabilized attosecond pulse trains with different carrier waves can be generated which may expands it application on the detection of ultra-fast dynamics processes.
In this paper, the HHG spectra of the helium atom driven by sawtooth-like fields have been investigated. The sawtooth shape parameter α has great influences on the coherence of the HHG spectrum. By adjusting the parameter α, not only the harmonic components, but also the relative intensity between the long and the short paths of the spectrum in the plateau region can be controlled. If proper α is chosen, the HHG spectrum is highly coherent in the plateau region from which phase-stabilized APT can be obtained with different center frequencies.
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