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We theoretically study high-order harmonic generation (HHG) and attosecond pulses from an atom irradiated synchronically by a terahertz (THz) pulse and an infrared laser pulse. For the HHG spectrum from the THz pulse alone and the combined pulse, an apparent peak-valley structure appears the platform region. Specially, for the periodic structure generated by an atom under the action of the combined pulse is originated from the interference between the electrons ionized at different instants in the laser field, which undergo many recollision and return to the core at the same time. Therefore, continuum harmonics with few chirps from the interference enhancement region can be achieved, which result in a chirp-free isolated attosecond pulse.
© 2015 Optical Society of America
1. Introduction
High-order harmonic emission can be generated from intense lasers interacting with atoms or molecules. For a typical spectrum of high-order harmonic generation (HHG), a platform with almost same intensity appears in a wide frequency range from tens to thousands of harmonic orders. Hence, HHG can be used to produce attosecond pulse, which is one of the most powerful tools to detect ultrafast electronic dynamics in atomic or molecular systems [1–3]. Recently, the maximum energy of harmonics from the interaction between a mid-infrared laser and a high-pressure noble gas has exceeded 1600 eV in experiment [4]. By using such broad harmonic spectrum, it is possible to obtain an ultrashort pulse with a duration as short as 2.5 attosecond in principle. However, the duration of the available shortest attocecond pulse in experiment are much longer than that [5]. One of the main limitations on getting shorter attosecond pulses is the existence of inherent chirp in the emission of high harmonics [6]. According to the three-step model, the ionized electrons with different energies come back to the core at different instants, which leads to unsynchronous emission times of harmonics [7, 8]. Therefore, the harmonic chirp compensation are usually adopted in experiments for producing ultrashort attosecond pulses from HHG. Many methods were employed to compensate the harmonic chirp, such as the chirp multilayer X-ray mirror, the negative group delay dispersion, and using the weak second harmonic laser [9–12]. However, these schemes can be effective only in a limited range of the harmonic emission. It is necessary to compensate the harmonic chirp in a wide frequency region for generating shorter attosecond pulses. In this paper, we propose a method for reducing the harmonic chirp in a single atom response by adding an infrared laser to a terahertz (THz) pulse.
At present, through optical rectification [13] and difference frequency amplification method in the experiment, the THz pulse’s intensity has reached 100 MV/cm with a carrier frequency up to 72 THz [14]. With such a strong THz pulse interacting with an atom and a molecule, the high harmonic emission has been observed and applied to produce ultrashort attosecond pulse. Hong [15] attempted to use a strong THz pulse interacting with an atom, and obtained an attosecond pulse train with a stable carrier phase. In order to obtain isolated attosecond pulse, people used a THz pulse combined with a high frequency laser pulse to act on an atom [16]. Through a molecular medium irradiated by a THz pulse synthesized by several linearly polarized laser pulses or a circularly polarized one [17, 18], isolated attosecond pulses can also be produced. In this work, we focus on harmonic generation mechanism from a THz pulse synchronizing an infrared laser pulse and isolated attosecond pulses by using the chirp free harmonic. It can be found that, the interaction between the combined pulse with a target gas, not only greatly improve the harmonic efficiency, but also reduce the harmonic chirp and obtain a much shorter attosecond pulse.
2. Theoretical method
In the length gauge and the dipolar approximation, we use the time-dependent Schrödinger equation (TDSE) to simulate the interaction between a atom and an intense laser pulse, and the TDSE can be described as follow (if no special instructions, atomic units are adopted throughout):
where is the soft-core potential describing the interaction between the electron and the nuclei, whose parameters are set as Z = 1, a = 0.4826, and the lowest bound energy is −0.9, which corresponds to the ground-state energy of the helium atom. E(t)x is the interaction between the atom and the laser electric field. The electric field of the synthesized two-color laser pulse is expressed as: Here, E1, E2, ω1, ω2, ϕ1, ϕ2 are the peak amplitude, the frequency, and the carrier-envelope phase of the two laser pulses, respectively. f1(t) = sin2(ω1t/τ1), f2(t) = cos2((ω2(t − tdelay))/2τ2) are the envelopes of the laser electric fields. Where tdelay = 7τ1/60 is the time delay of the infrared laser relative to the THz pulse. The time-dependent wave function at any instant can be obtained by solving the Equation (1) using the Crank-Nicholson method [19–21]. The time-dependent induced dipole acceleration can be given by Ehrenfest’s theorem: and the HHG spectrum can be determined by Fourier transforming the dipole acceleration a(t), . In order to investigate the dynamical process of harmonic emission, we perform the time-frequency analysis of the time-dependent induced dipole moment by means of the wavelet transform [22–24]: The core of the wavelet is selected as . The time-dependent ionization probability is written as: here, ci(t) = 〈φi|ψ(x, t)〉 is the population amplitude of the bound state φi.3. Results and discussions
We first study HHG from the atom exposed to a 50 fs THz pulse at 9120 nm alone and the two-color field synthesized by the THz pulse and a 10 fs infrared laser at 800 nm, the intensities of which are E1 = 0.02, E2 = 0.1, respectively. Fig. 1(a) shows the electric fields of the THz pulse (red solid curve) and the combined one (black dotted curve). The corresponding HHG spectra are presented by the red solid and black dotted curves in Fig. 1(b). It can be seen that, for the HHG spectrum from the THz pulse alone and the combined pulse, an apparent peak-valley structure appears the platform region. However, for the HHG spectrum from the combined pulse, the harmonic efficiency of which is enhanced about four orders of magnitude.
Fig. 1 (a) Electric fields of the THz pulse (red solid curve) and the combined-pulse (black dotted curve), (b) HHG spectra from the atom irradiated by the THz pulse (red solid curve) and the combined pulse (black dotted curve), the frequency unit is the fundamental frequency of the THz pulse.
To more clearly understand the enhancement of the harmonic emission, we calculate the time-dependent ionization probability and the ground state population. As shown by the black dotted and red solid curves in Fig. 2(a), the ionization probability of the combined pulse is larger than that of the single THz pulse because of the action of the additional infrared laser. The ground state populations in both cases are in the same magnitude, and the atom is not depleted, as depicted in Fig. 2(b). Because the harmonic emission is a process of a stimulated recombination[25], the harmonic efficiency is closely correlated with the probability amplitudes of the continuous and the ground states at the recombination instant. Therefore, the product of the ground state population and the continuum state amplitude in the combined pulse is larger than that of the THz pulse alone, which results in the great disparity in the harmonic intensity.
Fig. 2 The time dependence of the ionization probability (a) and the ground state population (b) in the THz pulse alone (red solid curve)and the combined laser (black dotted curve).
To clarify the peak-valley structure from the harmonic platform in the combined pulse, we perform the wavelet transform of the time-dependent induced dipole acceleration to observe the transient information of harmonic emission, as shown in Fig. 3. The overall structure of the time-frequency profile is shown in the subgraph. From the figure, one can see the harmonic emission is mainly focused on three regions. Firstly, the harmonic intensity near the 0.3 optical cycles is strong and the maximum harmonic order is lower than 1000th. Because the intensity of the THz pulse is very weak at this instant, the additional infrared laser mainly contribute to these harmonics. Secondly, the harmonic efficiency near the 1.5 optical cycles is weaker, and its cut-off is consistent with that from the single THz pulse. Hence, it can be inferred that these harmonics are greatly originated from the THz pulse. Finally, the harmonic emission near the 1.0 optical cycle has the highest cut-off and more intense efficiency. It can be noted that the harmonic emission exists obvious interference structure in this region. It is the interference that determines the appearance of peaks and valleys in the HHG spectrum. More importantly, the harmonic chirp in these interference regions is largely reduced. Hence, it is possible to generate shorter attosecond pulses from these chirp-free harmonics.
Fig. 3 The time-frequency analysis of the harmonics under the combined pulse and the classical simulation of the ionized electrons.
In order to investigate the origin of the harmonic interference, we also calculate the classical action of the ionized electrons [26]. Here, the initial spatial position and momentum are set as zero, and the effect of the atomic potential is ignored. One can obtain the variation of the ionized electron’s kinetic energy at the recombination instant on the emission times. It can be seen that, the electrons ionized at different instants come back to the core at the first (black filled circle), the second (white filled circle), the third (orange triangle), and the fourth (pink triangle) times. One can observe that the electrons ionized from different instants can simultaneously recombine to the parent ion with the same kinetic energy. The effect of multi-rescattering to the ion of the ionized electron are also investigated in the Dashcasan’s work [27]. It also can be found that the classical result presents an excellent agreement with the transient behavior of the harmonic emission from the quantum simulation.
Next, we take four trajectories (marked as A, B, C, and D) as example to study the interference process. These trajectories are originated from the electrons with different ionized instants returning to the parent ion at the same time 0.836 O.C., as shown by Fig. 4(a). It can be noticed that, the electron in the four cases recombines with the parent ion at this instant. However, for the trajectory D, the ionized electron collides with the core only once; for the trajectory A, the ionized electron has three collision with the core when it come back to the parent ion at the instant 0.836 O.C.. In terms of this classical model, we also calculate the kinetic energy of the ionized electron as a function of the time, as presented in Fig. 4(b). It can be observed that, the kinetic energy of the ionized electron at the instant 0.836 O.C. are almost the same for the above four trajectories, which all contribute to the same order harmonic. Therefore, the interference between different trajectories leads to the peak-valley spectral characteristics in the harmonic platform, as shown by the black dotted curve in Fig. 1(b).
Fig. 4 (a) The laser electric field (blue solid line) and the four classical trajectories, (b) the variation of the electron’s kinetic energy with the time.
To further confirm this interference behavior, we also study the time evolution of the electronic density distribution along the polarization direction of the THz pulse, as depicted in Fig. 5. One can see that, when the infrared laser is applied, the electron is mainly ionized at these instants, then oscillates in the THz laser field, and may collides with the core several times, which agree well with the corresponding classical trajectories.
Through the above analysis, one can find that the electronic wave packets ionized at different instants return to the parent ion at the same time, thereby the chirp of harmonics are largely reduced, as shown in Fig. 3. Hence, it is possible to obtain an ultrashort isolated attosecond pulse from these chirp-free harmonics. We also know that the phase effect of spectra plays an important role in the generation of ultrashort attosecond pulses. Therefore, we calculated the second derivative of a harmonic phase with harmonic order for the spectra generated by an atom under the action of the THz pulse or the combined pulse, as shown in Fig. 6. Compared with the THz pulse (red line), one can see that the second derivative of the harmonic phase with harmonic order in the combined pulse (black line) is near to 0 in the range from 1565th to 1624th harmonics(marked as the black pane in Fig. 6). Therefore, an isolated 196as pulse can be directly generated by synthesizing the harmonics from 1565th to 1624th harmonics without phase compensation (black line), as shown in the inset of Fig. 6. On the other hand, in the case of the single THz pulse, there are two adjacent attosecond pulses in every half optical cycle, and the duration of the shorter one is 247as, the intensity of which is greatly reduced in comparison with the case of the combined pulse. By changing the wavelength of the THz pulse or the time delay between the infrared laser and the THz one, the dynamic of the ionized electron can be controlled, which may result in shorter attoseond pulse with reduced chirp harmonics.
Fig. 6 Computation irradiated by the THz pulse (red solid curve) and the combined pulse (black dotted curve) shows the role of the second derivative of a harmonic phase with the harmonic order. Inset: Ultrashort pulse generated by synthesizing the harmonics from 1565th to 1624th from the single THz pulse (red dotted curve) and the combined-pulse (black solid curve).
4. Conclusion
In summary, we have proposed a method for the ultrashort isolated attosecond pulse generation from the chirp-free harmonics by a THz pulse in combination with an infrared laser. There exists many peaks and valleys in the harmonic platform. It is demonstrated that, these peaks are produced by the interference of the electrons ionized at different instants returning to the parent ion at the same time, which also apparently reduce the harmonic chirp. Therefore, ultrashort isolated attosecond pulse can be achieved by these chirp-free harmonics.
Acknowledgments
This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2013CB922200 and the National Natural Science Foundation of China (NSFC) (Grants No. 11274001, No. 11274141, No. 11304116, No. 11534004 and No. 11247024), and the Jilin Provincial Research Foundation for Basic Research, China (Grant Nos. 20130101012JC and 20140101168JC). J.G.C. acknowledges the support of the Academic Climbing Project of the Youth Discipline Leader of the Universities in Zhejiang Province under Grant No. pd2013415. We acknowledge the High Performance Computing Center of Jilin University for supercomputer time.
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