High-order harmonic generation (HHG) has been long investigated for its potential applications in modern physics.^[1–6] The basic understanding of the high harmonic emission can be provided by the three-step model.^[7–9] Electrons are ionized by tunneling to the continuum state; then the freed electrons move with oscillation of the electronic field and after the electric field changes the direction, the electrons may be captured by the parent ion again, radiating high energy harmonic photons. The harmonic spectrum is sensitive to the structure of molecules and induced dynamics, which makes HHG a promising tool to access ultrafast molecular imaging, observation of electron dynamics and molecular structures.^[10–12]

With the development of ultrafast laser pulse technology, HHG from molecules has been investigated both theoretically and experimentally.^[13–16] Studies on two-center system such as and H₂ showed that the nuclear motion can weaken the efficiency of molecular HHG.^[17,18] Spectral minimum induced by two-center interference in HHG from diatomic molecules such as N₂, O₂, CO also contains the information of molecular orbital and structures.^[19,20] For the three-center system, a rich set of physical phenomena has attracted a lot of attention. The orbital density imaging of H₂O has been realized by the calculation of the orientation-dependent ionization yield.^[21] The HHG of the equilateral was investigated in numerical solutions of the time-dependent Schrödinger equation (TDSE), and the results indicated that the harmonic emission largely depends on the laser–molecule orientation and internuclear distance.^[22] Qin et al.^[23] demonstrated that the interference of the HHG spectra from CO₂ at different alignment angles can modulate the spectra efficiently. It has also been shown that the HHG from CO₂ is very sensitive to multiple molecular orbitals that the effect of the highest occupied molecular orbital (HOMO) and HOMO-2 is important at small alignment angles and only HOMO becomes dominant at large alignment angles.^[24,25] Chen et al.^[26] revealed that the ionization process can remarkably influence the intersections of the HHG spectra from O₂ and CO₂ molecules at different orientation angles. Moreover, the ultrafast molecular imaging of CO₂ has become possible by electron diffraction^[27] or extracting the photoelectron momentum spectra of CO₂ probed with circularly polarized laser pulse.^[28] Some other polyatomic molecules have also begun to generate further interest. The harmonic yield on different angles between the molecular axis and the laser polarization has proved to contain the fingerprint of the highest occupied molecular orbitals in acetylene and allene, which agrees with the numerical results.^[29] Le et al.^[30] proposed an efficient method of the quantitative rescattering theory (QRS) for calculation of HHG from CCl₄ molecule, which well reproduces the minimum in HHG spectra observed in available experiments. It has also been reported that the HHG intensity can be enhanced with armchair grapheme since the initial electron density distribution along the x axis is much larger than that along the y axis.^[31] Very recently, harmonic emission in benzene with linearly and circularly polarized laser pulses has illustrated that the ionic motion has a large effect on the harmonic response.^[32]

Here we demonstrate the HHG of CO₂ with different vibrational modes (balance vibration, bending vibration, stretching vibration) driven by an intense laser field. The results show that the high harmonic efficiency of CO₂ with stretching vibrational mode is the strongest. The underlying physical mechanism of the HHG can be well explained by the corresponding ionization yield and the time–frequency analysis. When the CO₂ molecule with stretching vibrational mode is exposed in the intense laser field with τ_d = 3 fs, a strong and single attosecond (asec) pulse can be generated.

The calculations performed in this paper are based on the strong-field Lewenstein model, which has been widely used to investigate the process of harmonic emission.^[7,33,34] The transition amplitude has the form of

The harmonic spectrum intensity is proportional to the Fourier transformation of the time-dependent dipole acceleration

By superposing several orders of the harmonic spectrum (here q is the harmonic order) we can obtain an attosecond pulse as follows:

The highest occupied molecular orbital (HOMO) of CO₂ molecule is constructed by the GAMESS-UK package,^[35] which is used as the initial input. In the gas phase, the CO₂ molecule has three internal vibrations: the balance vibrational mode (the atoms symmetrically stretch along the molecular axis), bending vibrational mode, and stretching vibrational mode (the atoms asymmetrically stretch along the molecular axis). The three different vibrational modes are taken into account and the corresponding electron density of coordinate–space orbital |φ(r)|² is shown in Fig. 1. The density distribution is very sensitive to the molecular vibrational modes and shows different characteristics depending on different vibrational modes. We can see from Figs. 1(a)–1(c), the electron density distribution of the balance vibrational mode has a “mirror symmetry” structure, which is broken in bending and stretching vibrational modes and gradually converges.

Fig. 1.

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	Fig. 1. The electron density of coordinate–space orbital |φ(r)|2 of the HOMO of CO2 molecule with (a) balance vibrational mode, (b) bending vibrational mode and (c) stretching vibrational mode.

We apply the strong-field Lewenstein model to demonstrate the HHG of CO₂ in a linearly polarized laser field, which has the form as E(t) = E₀f(t)cos(ωt), where is the Gaussian envelope; the peak intensity of the laser pulse is chosen as 6 × 10¹⁴ W/cm². The frequency ω = 0.0569 a.u. (800 nm in wavelength) and τ_d is the full-width at half-maximum (FWHM) of the laser field. τ_d = 3 fs and τ_d = 5 fs are considered respectively. The lasers’ polarization directions are assumed to be parallel to the molecular axis in our calculations and the corresponding electric fields are shown in Fig. 2.

Fig. 2.

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	Fig. 2. The electric fields of the 800 nm laser pulse (red solid line) and corresponding pulse envelope (black dashed line) with (a) τd = 3 fs, (b) τd = 5 fs.

Figure 3 shows the HHG of the CO₂ molecule with different molecular vibrational modes for τ_d = 3 fs and τ_d = 5 fs. For τ_d = 3 fs, figure 3(a) shows that the harmonic intensity of the CO₂ with the stretching vibrational mode is at least 3 orders of magnitude higher than that with the balance vibrational mode, and the intensity of the spectrum with the bending vibrational mode is a little stronger than that with the balance vibrational mode. These results indicate that the HHG spectrum is sensitive to molecular vibrational modes.

Fig. 3.

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	Fig. 3. The HHG of CO2 with different vibrational modes in the intense laser field with the peak intensity of 6 × 1014 W/cm2. (a) τd = 3 fs, (b) τd = 5 fs.

For τ_d = 5 fs, figure 3(b) shows that the spectrums with three vibrational modes exhibit the similar tendency except more modulations compared with that for τ_d = 3 fs.

Figure 4 shows the ionization yield of CO₂ molecule with different vibrational modes for the cases of τ_d = 3 fs and τ_d = 5 fs. It is clear to see that the ionization yield of CO₂ with the balance vibrational mode is the lowest, the ionization yield of CO₂ with the bending vibrational mode is a little enhanced and the ionization yield of CO₂ with the stretching vibrational mode is the largest, which results in the enhancement of HHG intensity with the stretching vibrational mode. We can find from Fig. 1 that the electron density distribution of the balance vibrational mode distributes around the two carbon nuclei and the electron density distribution of the stretching vibrational mode almost distributes around the carbon nucleus in the positive z direction. Electrons are attracted by two carbon nuclei in the balance vibrational mode, but in the stretching vibrational mode electrons feel the attractive forces mostly from one carbon nucleus, which may lead to the electrons being easy to ionize in the stretching vibrational mode.

Fig. 4.

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	Fig. 4. Ionization yields of CO2 molecule with different vibrational modes. (a) τd = 3 fs, (b) τd = 5 fs.

We investigate the time frequency distributions of harmonics from the 60th to 90th of CO₂ with different vibrational modes for the cases of τ_d = 3 fs, which are shown in Figs. 5(a)–5(c). It is clear to see that the intensity of the emission peaks with the stretching vibrational mode are the strongest, which is in accordance with the high harmonic generation and the corresponding ionization yields shown in Fig. 3(a) and Fig. 4(a).

Fig. 5.

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	Fig. 5. (a)–(c) Time frequency distributions of harmonics from 60th to 90th and (d)–(f) the temporal profiles of generated attosecond pulses from CO2 with different vibrational modes for the case of τd = 3 fs.

Figures 5(d)–5(f) show the temporal profiles of the generated attosecond pulses from CO₂ by performing the inverse Fourier transforms of the spectrum with different vibrational modes. Figure 5(d) shows that, by properly filtering the harmonic spectrum from 54th to 82th with the balance vibrational mode, an isolated attosecond pulse with a duration of about 120 as is generated. An asec pulse train with two radiation pulses is generated with bending vibrational mode as depicted in figure 5(e) by superposing harmonics from the 61th to 83th order. Figure 5(f) shows that a strong and isolated asec pulse with a duration of about 112 as is obtained with the stretching vibrational mode by superposing the harmonics from the 57th to 83th order. Furthermore, the peak intensity of the asec pulse is gradually enhanced from Fig. 5(d) to Fig. 5(e), and the intensity of asec pulse with the stretching vibrational mode is about 3 orders of magnitude higher than that with the balance vibrational mode.

We demonstrate the generation of high-order harmonic and attosecond pulse of CO₂ with different vibrational modes by the strong-field Lewenstein model. The results show that the high-order harmonic yield is sensitive to the molecular vibrational mode, and the efficiency of harmonic spectrum from CO₂ with the stretching vibrational mode is the strongest. The underlying physical mechanism of the HHG can be well explained by the corresponding ionization yields and the time-frequency analysis. When the CO₂ molecule with the stretching vibrational mode is exposed in the intense laser field with τ_d = 3 fs, a strong isolated attosecond pulse with a duration of about 112 as is generated.