1. Introduction

In the semiclassical three-step model of high-order harmonic generation (HHG), active electrons first tunnel through the potential barrier, are then accelerated by laser fields, and finally recombine with parent ions to emit high-energy photons [1–2]. This process can provide the approach to probe electronic and nuclear dynamics of atoms and molecules with an unprecedented fast temporal resolution [3–6]. For example, recently, Wörner et al. [7] followed a chemical reaction of less than an attosecond between the stretched and compressed geometry of weakly vibrationally excited Br₂ using high-harmonic interferometry, and Goulielmakis et al. [8] probed the real-time valence electron motion through the spectrally resolved absorption of an attosecond extreme-ultraviolet pulse.

The harmonic emission from impulsively aligned CO₂ shows a special feature that the harmonic yield in some spectral region exhibits an inverted modulation versus the molecular alignment parameter, and the modulation inversion is attributed to the interference of recombining electron from the two oxygen atoms [9–16]. Boutu et al. [17] even gave the phase measurement of harmonics generated in aligned CO₂ molecules, which provided proof for an interpretation in terms of destructive quantum interference. However, the harmonic orders that are inversely modulated differed in different experiments from CO₂ molecules, and the effect of the molecular potential on the recolliding electron wave packet was treated differently in the interpretation of experimental data [9–13]. Therefore, we attributed such phenomena to the laser field distortion of the molecular HOMO, which should play a key role in such quantum system [16]. Moreover, the role of the multiple HOMOs effect was also proposed for the laser intensity dependence in aligned molecules recently [18–23]. Probably the field distortion of the molecular HOMO and the multiple HOMOs effect both play roles in such system. We would like to point out that, among the explanations for the laser intensity dependence on the modulation inversion of the harmonic yield, our model focuses on the field distortion effect on the molecular HOMO and has the merit of simplicity. In this letter, we further represent the new experimental results on O₂ molecules which can be well explained by our model.

The previous results [9–10] [12–23] about the interference (or the laser intensity dependence) mainly focus on CO₂ and N₂ molecules, and there is a comparative lack of the investigation on O₂ molecule. Infrequently for O₂ molecule, Itatani et al. [3] have showed that the high harmonics are enhanced when aligned near 45° with respect to the laser polarization due to the π_g symmetry of its HOMO, and Vozzi et al. [11] reported the order dependence of the harmonic yield due to the two-center interference. Here we further measure the angular distribution and the laser intensity dependence of the harmonic yield from O₂ molecules.

In this work, we investigate the two-center interference in aligned O₂ molecules and perform a more comprehensive study of the interference characteristics by comparing with CO₂ molecules. Through the measurement of both the temporal evolution and the angular distribution of harmonic intensity, the detailed characteristics of intensity enhancement and suppression are observed, and the effects of the laser intensity and the two-centre interference are summarized. Moreover, the shift of the constructive spectral region in aligned O₂ molecules is demonstrated by tuning the driving laser intensity.

2. Theoretical model and experimental setup

2.1 Theoretical model

For the high-order harmonic emission from impulsively aligned O₂ or CO₂ molecules irradiated by an ultra short laser pulse, the quantum interference is attributed to the interference of recombining electron from the two oxygen atoms [10–11]. For such antisymmetric atomic orbitals of the linear molecules, the interference factor is defined as$\left[1-\cos (2\pi R\cos \theta /{\lambda }_B)\right]$, and the harmonic ratio S/S₀ between the signal from aligned molecules and that from an isotropic ensemble, at the considered time delay t ₀, can be simply approximated as follows:

where θ is the angle between the laser polarization and the internuclear axis, P_Θ(θ, t₀) [10, 11, 15] is the angle distribution of the molecular ensemble at time t₀, probed at an angle Θ from the aligning field direction; n is the harmonic order, R is the internuclear separation, C is a fitting constant; λ _B(n) is the de Broglie wavelength of an electron responsible for the emission of the nth harmonic, given here by ${\lambda }_B(n)=h/\sqrt{2m_e{E}_k}$, where m _e is the electron mass, E_k is the electron kinetic energy. Considering the field distortion of the molecular potential, we treat the kinetic energy of the returning electron as $E_k=nh{v}_0-\delta I_p$ (δ≥0, variable) [16], where nhv ₀ is the emitted photon energy (hv ₀=1.55eV), and $\delta I_p$is the effective molecular potential which is closely related to the laser field. The effective ionization potentials could also be expanded to the new understanding of the continuum electron wave functions, and the field-free scattering states are now known to give good agreement with experiments [23] [25].

In this model, we directly and simply convolve the molecular distribution with the interference intensity$\left[1-\cos (2\pi R\cos \theta /{\lambda }_B)\right]$ for its simplification, and this is a rather crude approximation and a strongly simplified model to give a qualitative explanation. The laser field distortion of the molecular HOMO should play a key role through the effective molecular potential. The order dependence is shown as a function of the harmonic order n; the angle dependence is shown as a function of the pump-probe polarization angle Θ; and the laser intensity dependence is shown as the effective molecular potential$\delta I_p$ through by${\lambda }_B(n)=h/\sqrt{2m_e(nh{\nu }_0-\delta I_p)}$.

2.2 Experimental setup

The experiments are performed by using a Ti:sapphire-based chirped pulse amplification laser system (Spectra-Physics, TSA-25), which produces 50 fs laser pulses at 800 nm center wavelength with a repetition rate of 10 Hz. The laser pulse is split into two beams, one used as the pump pulse (for aligning molecules) and the other as the probe pulse (for driving HHG from the molecules). The two beams are collinearly focused with variable pump-probe time delay on a pulsed supersonic molecular beam located in a high-vacuum interaction chamber. The aligning laser energy is kept constant and the intensity is estimated to be 4 × 10¹³ W/cm² at the interaction region within the gas jet. The probe laser energy is adjustable by using a half-wave plate and a high-extinction film polarizer. The stagnation pressure of CO₂ gas is around 2 bars, leading to a rotational temperature of tens of kelvins. The generated high-order harmonics are detected by a home-made flat-field grating spectrometer equipped with a soft X-ray charge-coupled device camera (CCD, Princeton Instruments, SX 400).

The experimentally measured 23rd harmonic ratio S/S₀, with (S) and without (S₀) the aligning laser pulses, as functions of pump-probe delay are represented in Fig. 1 (the black solid lines and left axis, the probe laser intensity is 2.8 × 10¹⁴ W/cm², the polarizations of the aligning and probe pulse are parallel). The calculated time-dependent alignment parameters $<{\cos }^2\theta >(t)$ [15–16] are also shown in Fig. 1 (the red dashed curves and right axis, the initial rotational temperature is taken to be 80 K). It is evident that the modulation of harmonic signal for CO₂ molecules is reversely matched with that of the molecular alignment parameter$<{\cos }^2\theta >$, while the modulation of harmonic signal for O₂ molecules is matched with the molecular alignment parameter $<{\cos }^2\theta >$ exactly. The phenomena are regarded as the evidence of interference of the recombining electron originated from two oxygen atoms [10–11].

 

Fig. 1 Measured 23rd harmonic ratio S/S₀ as functions of pump-probe delay in aligned (a) CO₂ and (b) O₂ molecules. The dashed lines show the calculated alignment parameters$<{\cos }^2\theta >$ as functions of the pump-probe delay.

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3. Results and discussion

3.1 Experimental results

For the detailed observation of the interference characteristics from different harmonic orders, we present the harmonic ratio S/S₀ at the half and quarter revival from three different harmonic orders, which are shown in Fig. 2 together with the calculated time-dependent alignment parameters$<{\cos }^2\theta >$. For O₂ molecules shown in Figs. 2 (a) - (c), the results are measured at the driving laser intensity of 2.8 × 10¹⁴ W/cm² for 21st, 29th, and 37th harmonic orders, and they display that the constructive interference gradually weakens and even inverses from 21st to 37th harmonic order. For CO₂ molecules shown in Figs. 2 (d) - (f), the results are measured at the driving laser intensity of 2.0 × 10¹⁴ W/cm² for 21st, 25th, and 29th harmonic orders, and they display that the destructive interference gradually weakens and even inverses from 21st to 29th harmonic order. Both the two gases display the order dependence of the two-center interference.

 

Fig. 2 Measured temporal evolutions of the harmonic ratio S/S₀ at the half and quarter revival from aligned O₂ molecules for (a) 21st, (b) 29th, and (c) 37th harmonic orders, and from aligned CO₂ molecules for (d) 21st, (e) 25th, and (f) 29th harmonic orders. The dashed curves are the calculated alignment parameters$<{\cos }^2\theta >$.

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We also measure the angular distributions of HHG as functions of the pump-probe polarization angle (the polarization of the aligning pulse is fixed) at the top alignment of the half revival, and the results are shown in Fig. 3 . Due to the symmetry, we only measure the angle in 0° – 90° and make the polar plot. The angular distributions of the 19th (red star) and 27th (black dot) harmonic orders from aligned O₂ molecules are shown in Fig. 3(a), and the same distributions from aligned CO₂ molecules are shown in Fig. 3(b). The driving laser intensities for both the two gases are estimated to be 2.8 × 10¹⁴ W/cm². These phenomena of the two gases at the certain laser intensity are similar to the observation of Mairesse et al. [24], showing that the harmonic emission of O₂ molecules is maximized at 0° while the emission of CO₂ molecules is maximized at 90°. The simulated results fitted with Eq. (1) are also shown in Fig. 3(c) for O₂ molecules and in Fig. 3(d) for CO₂ molecules. The simulated results in Fig. 3(c) for O₂ molecules are using a slightly longer R = 0.133nm and $\delta I_p$ = 0eV, and the simulated results in Fig. 3 (d) for CO₂ molecules are using a slightly longer R = 0.245nm and $\delta I_p$ = 13.77eV. These figures show that the simulations can well reproduce the experimental results, and the angular distributions are dependent on the harmonic orders, which result from the two-center interference.

 

Fig. 3 Measured angular distributions of the harmonic ratio S/S₀ as functions of the pump-probe polarization angle Θ at the top alignment of the half revival from (a) O₂ molecules and (b) CO₂ molecules. The simulated results are fitted with Eq. (1) for (c) O₂ molecules and (d) CO₂ molecules.

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The order dependence of the characteristic enhancement and suppression are obvious in both gases through the temporal evolution (shown in Fig. 2) and the angular distribution (shown in Fig. 3). Due to that the internuclear separation of CO₂ molecule is much longer than that of O₂ molecule, the destructive interference first appears at the 21st harmonic order of CO₂ molecules, and then appears at the 37th harmonic order of O₂ molecules in our observation window.

We then further investigate the role of the laser intensity, which can shift the spectral region of the two-center interference and can inverse the modulation of the harmonic emission. For O₂ molecules shown in Fig. 4(a) , we measure the experimental spectral distributions of the harmonic ratio S/S₀ with three different driving laser intensities at Θ = 0°, together with the calculations by numerically solving Eq. (1) using the internuclear separation R = 0.121nm (the solid curves), 0.133nm (a slightly longer R, the dashed curves) and the effective molecular potential $\delta I_p$ = 0eV (the thick curves), 12.1eV (the thin curves). We can observe that the spectral region of the constructive enhancement (S/S₀ > 1) in aligned O₂ molecules can be shifted to the higher harmonic order by enhancing the driving laser intensity: the constructive enhancement appears for the 19th - 23rd harmonics at the driving laser intensity of 1.8 × 10¹⁴ W/cm² (circles), for the 19th - 29th harmonics at 2.3 × 10¹⁴ W/cm² (squares), and for the 19th – 31st harmonics at 2.8 × 10¹⁴ W/cm² (triangles). However, the destructive suppression (S/S₀ < 1) in aligned O₂ molecules should occur at the higher orders of harmonics, which need the much higher intensity of the driving laser field. At the maximal driving laser intensity (2.8 × 10¹⁴ W/cm²) under our experimental condition, the destructive suppression (S/S₀ < 1) in aligned O₂ molecules appears for the 33rd-37th harmonic orders, and it is different from the observation of Vozzi et al. [11] that the suppression appears for the 41st–47th harmonic orders at the driving laser intensity of 3.5 × 10¹⁴ W/cm². Furthermore, our experimental results of O₂ molecules are best fitted by the electron kinetic energy as $E_k=nh{v}_0$ (the thick blue curve), while the results of Vozzi et al. are more fitted by the electron kinetic energy as $E_k=nh{v}_0-I_p$ (the red thin curve) at the comparative high laser intensity. These results of O₂ molecules show that the higher driving laser intensity corresponds to the larger effective molecular potential. As a comparison, we also plot the spectral distributions in aligned CO₂ molecules at the different driving laser intensity, shown in Fig. 4 (b) [16]. They also show that the spectral region of the destructive suppression (S/S₀ < 1) can be shifted to the higher harmonic order by enhancing the driving laser intensity. Therefore, the results of both two gases indicate that the two-center interference does not follow a relationship corresponding to a fixed molecular potential when the driving laser intensity is varied, which can be explained by our simple model.

 

Fig. 4 Measured spectral distributions of the harmonic ratio S/S₀ at three different driving laser intensities of 1.8 × 10¹⁴ (circles), 2.3 × 10¹⁴ (squares) and 2.8 × 10¹⁴ (triangles) W/cm², (a) from aligned O₂ molecules with the pump-probe delay 5.7ps, and (b) from aligned CO₂ molecules with the pump-probe delay 21.1ps. The curves are fitted with Eq. (1) using the internuclear separation R and δI_p = 0 (blue thick solid line), δI_p = I_p (red thin solid line); also using a slightly longer R(dashed lines).

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3.2 Discussion

We use the slightly longer R in simulations for two reasons: (1) the field distortion of the molecular HOMO cannot only change the effective molecular potential$\delta I_p$, but also stretch the effective distance R between the centers of electron density; (2) If $\delta I_p$is restricted to being 0, the simulation using the slightly longer R more fits the experimental data, although the same effect can be reached by further decreasing $\delta I_p$<0.

There is almost ‘no destructive effect’ visible in aligned O₂ molecules due to that: (1) the destructive effect visible should appear at the much higher harmonic order, therefore one can’t see the effect visible at the low harmonic orders with decreasing intensity for O₂ molecules; (2) learning from Fig. 2, the Signal Noise Ratio from O₂ molecules is much smaller than that from CO₂ molecules, due to that it’s harder to get an impulsive alignment for O₂ molecules. Therefore, if decreasing the driving laser intensity, the Signal Noise Ratio is too small to identify the Signal under our experimental condition.

The dispersion relation could also be written as$E_k=nh{v}_0-I_p+{}^{*}I{}_p$, where $I_p$ is the original molecular potential, and ${}^{*}I{}_p$is the new active molecular potential considering the field distortion. If the driving laser intensity is very low and the field distortion of HOMO can be neglected (i.e. $I_p={}^{*}I{}_p$), the dispersion relation will be close to$E_k=nh{v}_0$ (i.e. $\delta I_p=I_p-{}^{*}I{}_p=0$). This is consistent with the experimental observation that the lower driving laser intensity is close to δI_p = 0. Therefore, $\delta I_p=I_p-{}^{*}I{}_p$ indicates the depth of the field distortion of the molecular potential, which should have a close relation to the driving laser intensity (i.e. they have the same trend —— the higher driving laser intensity, the larger depth$\delta I_p$). Whether $\delta I_p$>I_p or $\delta I_p$<0, we think they are possible at the extreme laser condition.

The experimental and calculated results of these two kinds of the π_g-type molecules indicate that the general feature of modulation inversion is governed by the interference effect, but the spectral region of modulation inversion can be shifted by tuning the driving laser intensity. Therefore, the two-center interference in the recombination process can be manipulated and even inverted from enhancement to suppression by changing the laser intensity through control over the effective molecular potential. The actual physical effect is that the higher driving laser intensity, the deeper field distortion of the molecular HOMO (The magnitude of the coefficient δ shows the depth of the field distortion of the molecular HOMO). These observations are helpful for the future experiment on such molecular system and important for the theoretical study on this topic.

Recently, through varying the wavelength and intensity of the generating laser field, Wörner et al. [23] have showed that the minimum in aligned N₂ molecules is nearly unaffected, whereas the minimum in aligned CO₂ molecules shifts over more than 15eV. The minimum in aligned N₂ molecules is a result of the orbital structure, whereas the minimum in aligned CO₂ molecules is dynamical and results from the interference between the multiple orbitals. These results directly present that the laser intensity dependence in aligned N₂ molecules is different from the laser intensity dependence in aligned π_g-type molecules.

4. Conclusions

In summary, we investigate the processes of the two-center interference in aligned O₂ molecules by comparing with CO₂ molecules, and observe the detailed characteristic enhancement and suppression in the measurement of the temporal evolution and the angular distribution of the harmonic intensity. For the both two gases, the angular distribution and the laser intensity dependence can be well explained by the modified interference model. Moreover, we demonstrate that the spectral region of the constructive enhancement in aligned O₂ molecules can be shifted by tuning the driving laser intensity, which is also supported by the modified interference model. Finally, the effects of quantum interference and laser intensity on HHG from the π_g-type molecules are summarized.