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High-order harmonic generation (HHG) in molecular targets is experimentally investigated in order to reveal the role of the nuclear motion played in the harmonic generation process. An obvious broadening in the harmonic spectrum from the H2 molecule is observed in comparison with the harmonic spectrum generated from other molecules with relatively heavy nuclei. We also find that the harmonic yield from the H2 molecule is much weaker than the yield from those gas targets with the similar ionization potentials, such as Ar atom and N2 molecule. The yield suppression and the spectrum broadening of HHG can be attributed to the vibrational motion of nuclear induced by the driving laser pulse. Moreover, the one-dimensional (1D) time-dependent Schrödinger equation (TDSE) with the non-Born-Oppenheimer (NBO) treatment is numerically solved to provide a theoretical support to our explanation.
© 2016 Optical Society of America
1. Introduction
High-order harmonic generation (HHG) [1–4] has been already proved to be a promising approach to track the electron dynamics in atoms and molecules [5,6], due to this kind of extreme nonlinear process occurring on an attosecond time scale. For atomic HHG, the semi-classical three-step picture [7,8] and the strong-field approximation (SFA) [9], usually called as the Lewenstein model, are used to describe the underlying mechanism of HHG. The molecular HHG can be described by the adapted SFA [10] in combination of the MO-ADK model [11]. Because the MO-ADK model is strongly dependent on the choice of the basis, the molecular will suffer from the same issue. The earlier investigation of the molecular HHG using the above models is implemented at the fixed internuclear distance within the framework of the Born-Oppenheimer approximation (BOA), which ignores the couplings of the electronic and nuclear wave packet. Recently, the direct solution of the time-dependent Schrödinger equation (TDSE) beyond the BOA has revealed a variety of unique nonlinear responses, including the charge-resonance-enhanced ionization (CREI) at large internuclear distance [12,13], even harmonic generation [14], an isolated attosecond pulse generation [15], the shortening of trains of the generated attosecond pulses [16], and the softening of the molecular bond by multiphoton couplings [17]. In addition, the monitoring of electron-nuclear dynamics on attosecond time scale has been investigated extensively to give a view on the effect of nuclear motion in HHG process [18–26]. These above phenomena can be attributed to the additional degree of freedom for nuclear wave packet in molecules. Various investigations of the influence of nuclear motion on molecular HHG process have been performed in H2+, D2+ or H2 [27–31]. Especially, a scheme reported in [30] is proposed to probe electron-nuclei correlation in frequency modulation by observing the redshift in molecular HHG. The experimental investigation of the nuclear motion effect on the HHG with the particular emphasis on the spectrum broadening is still very rare, although the HHG in molecules has been used to probe nuclear dynamics and structural rearrangement on an attosecond time scale [19–21].
In this work, we experimentally investigate the effect of nuclear motion on the molecular HHG under different experimental conditions, upon changing the focus position, varying the medium length and using different medium species for comparison. This experimental comparison includes two aspects: (i) we compare the harmonic yield among H2, Ar, and N2, due to the three gases having the similar ionization potential. It is found that the harmonic yield from H2 molecule is much weaker, especially in the case of the longer medium length. (ii) We compare the shape of high harmonic spectrum of H2 molecule with other gases having the relatively heavy nuclei (such as N2, O2, and CO2 molecules), and find the more broadening of the harmonic spectrum generated from H2 molecule, especially in the case of the longer medium length. The yield suppression and the spectrum broadening of HHG can be attributed to the vibrational motion of nuclear induced by the driving laser pulse. Moreover, the TDSE simulation considering the nuclear motion is performed to provide a theoretical support to such explanation.
2. Experiment
In our experiment, a commercial Ti: sapphire femtosecond laser system (Coherent, Inc) operating at 800nm wavelength with 45fs (FWHM) pulse duration at a repetition rate of 1kHz is used to produce the driving pulse. High-order harmonics are generated by focusing the 2mJ driving pulse into a gas cell in the high-vacuum interaction chamber. The focal length is about 400mm. The focus position with respect to the center of gas cell can be moved along the driving pulse propagating direction from Z = −8mm to Z = + 8mm Here, the negative (positive) Z value means that the driving pulse is focused after (before) the center of gas cell. The peak intensity of the driving pulse at the focus position (Z = 0mm) is about 2.5 × 1014W/cm2, which is estimated from the cutoff energy of the measured HHG. The Rayleigh length is measured to be about 8mm, i.e., from Z = −4mm and Z = + 4mm, therefore, the laser intensity in the interaction region at these two points (Z = +/− 4mm) can be obtained by dividing the peak intensity by a factor of 2. The laser intensities at other positions can be simply obtained according to the spatial distribution of the Gaussian beam, therefore, the axis of the laser intensity is nonlinear. Then, the generated harmonic signal is detected by a home-made flat-field soft X-ray grating spectrograph equipped with a soft X-ray CCD (Princeton Instruments, SX400). The spectrograph consists of a gold-coated spherical mirror, a gold-coated cylindrical mirror, a slit, and a Hitachi flat-field grating (1200grooves/mm). The spherical mirror acts as a spatial integrator in horizontal direction, the cylindrical mirror acts as an imaging mirror in vertical direction, and the slit with a width of about 0.2mm acts as a spatial window in vertical direction. A 500nm thick aluminum foil is used in the spectrometer to block the driving pulse. A diaphragm in front of the lens is used to further improve the driving pulse beam quality. We first optimize the high-order harmonic signal by adjusting the gas pressure and find that the measured harmonic yields are first increased and then decreased for all the gas species when it is adjusted from 10 to 50Torr. We select a standard of 20Torr in our experiments because this value is the approximate pressure for all of gas species to obtain the maximum harmonic yield.
3. Experimental results and discussion
Figures 1(a)-1(c) show the harmonic spectrum measured as a function of the focus position of the driving pulse for 1-mm long H2, N2, and Ar gas cell. The three harmonic spectra exhibit the similar cut-off order for the reason of the almost equal ionization potential of these above three species. However, the harmonic intensity and the spectral shape for these above three gases are very different. As shown in Figs. 1(d) and 1(e), the order of the measured harmonic intensities is H2<N2<Ar. Intuitively, the biggest difference of the atomic HHG is the absence of the nuclear motion which is the additional degree of freedom in diatomic molecules. As described in [20, 21, 23, 30], one has to include the nuclear motion in molecular HHG especially for light molecules, which can explain the lowered harmonic intensity from H2. Moreover, the mass ratio between nucleus and electron of N2 molecule is much bigger than the case of H2 molecule. By comparing H2 with N2, the harmonic yield is greater in the case of N2 whose nuclei are expected to move more slowly. For the above reasons, the contribution of the nuclear motion can be neglected for N2 and cannot be neglected for H2 in the molecular HHG process [21]. In addition, the recombination cross sections are different for atoms and molecules with different orbitals, which can lead to different HHG intensities by influencing the last step in HHG process. For example, the N2 molecule has an enhanced recombination cross section around 30 eV because of a shape resonance and a suppressed cross section due to a Cooper minimum for higher energies, while the Ar atom has a suppressed cross section (but no enhancement) due to a Cooper minimum around 53eV. This Cooper minimum in Ar is also the reason why the harmonics in Ar are being suppressed beyond the 29th harmonic order, as shown in Figs. 1(c) and 2(c). Lastly, the conditions of phase matching could be different among these three gases, because the phases of dipole are different due to the different interaction process between the ion and the electron. The slight differences in the ionization potential may alter the phase matching conditions [21], and the variation of phase matching may be one of the reasons of the order of the measured harmonic intensities from these three gases.
Fig. 1 Experimentally measured harmonic yield from (a) H2, (b) N2 and (c) Ar as a function of the focus position of the driving pulse in 1-mm long gas cell. The comparison of the harmonic yield in 1-mm long gas cell at the focus positions of (d) 0mm and (e) −6mm, which is extracted from (a), (b) and (c).
Fig. 2 Experimentally measured harmonic yield from (a) H2, (b) N2 and (c) Ar as a function of the focus position of the fundamental driving laser in 4-mm long gas cell. The comparison of the harmonic yield in 4-mm long gas cell at the focus positions of (d) 0mm and (e) −6mm, which is extracted from (a), (b) and (c).
In order to observe the characteristics of molecular HHG resulted from the nuclear motion effect, the harmonic spectrum as a function of the focus position of the driving pulse for a 4-mm long gas cell have been also measured, as shown in Figs. 2(a)-2(c). After increasing the length of medium, the macroscopic propagation effect cannot be neglected any more. The variation of the harmonic intensity and the harmonic spectrum shape occurs in long gas cell compared with the short gas cell. The experimental results show that the propagation effect can amplify the differences of HHG signal between H2 and the other two species. Both Figs. 1 and 2 show that an obvious blueshift is observed at the focus position of 0mm. One reason is that the ionized electrons can change the refractive index of the medium leading to the blueshift of the harmonic spectrum [30, 32]. Another reason is that the harmonic signal on the red side of the spectrum is reduced because the medium is already depleted on the trailing edge of the driving pulse [30, 33]. It is a good understanding that higher laser intensity at the focus position of 0 mm can ionize more electrons leading to the obvious blueshift. In addition, an obvious spectrum broadening of the H2 harmonic yield can be achieved at the focus position of about −7mm, especially in the case of long gas medium. In order to get direct insight on the spectrum broadening, we pay attention to the experimental macroscopic evolution of the harmonic spectrum as following.
The normalized harmonic intensities from H2 and N2 at three different focus positions ( + 1mm, −3mm and −7mm) in a 4-mm long gas cell are shown in Figs. 3(a) and 3(b). The harmonic spectrum shifts from the blue to the red with changing the focus position (corresponding to decrease the effective intensity of the driving pulse in gas medium). The observed spectrum shift can be attributed to the asymmetry of ionization signals [30]. The macroscopic evolution of the 21st order harmonics for the different medium length at the focus position of about −7mm has been also observed in order to research the influence of the propagation on the harmonic spectrum broadening, as shown in Figs. 3(c)-3(e). In the short length medium, the structures of the harmonic spectrum for all species used in our experiment (H2, N2, O2, CO2 and Ar) have similar asymmetric shapes. Compared to the short length medium, the spectrum structure gradually becomes different between H2 and other species with increasing the medium length. In addition to H2, the harmonic spectrum narrowing of other species can be explained in terms of the destruction of the relative phase of trajectories by the free electron dispersion with increasing the medium length. In the case of H2, considering the non-ignorable nuclear motion, the spectrum broadening could be explained according to the redshift induced by the asymmetry of ionization signals and the blueshift induced by the enhanced ionization induced by the vibrational nuclei contribute together to the spectrum broadening. As described in [30], the blueshift induced on the rising part of the pulse is comparable to the redshift on the falling part when the driving pulse intensity is below the ionization saturation threshold. In addition to the spectral broadening, there is a double peak structure of the harmonic spectrum from N2 in Fig. 3(c). The double peak structure probably originates from the contribution of the long and the short path, due to the short gas cell. In this case, the phase-matching effect is not sufficient to suppress the long path contribution.
Fig. 3 Experimentally measured harmonic spectrum from (a) H2 and (b) N2 for different focus positions in 4-mm long gas cell. The harmonic intensities of the 21st order from different gases (H2, N2, O2, CO2 and Ar) using gas cell of different lengths ((c) 1mm, (d) 2mm, and (e) 4mm) at the focus position of about −7mm.
3. Numerical simulation and discussion
In order to interpret the main feature exhibited by the experimental observation, we investigate the interaction of the H2 molecule with the driving pulse by numerical solution of the time-dependent Schrödinger equation (TDSE). The H2 molecule is simplified as a one-dimensional system, where the dynamics of two electrons and the nuclei are restricted to the laser polarization direction via two electron coordinates (z1, z2) and one nuclear coordinate R. This treatment can largely reduce the computation effort compared with the fully three-dimensional molecule, and meanwhile capture the essential of the electron-nuclei coupling beyond the Born-Oppenheimer approximation. After separating the center-of-mass motion, the TDSE for the internal motion reads [28]
where He is the Hamiltonian of the two electrons, HN is the Hamiltonian of the two nuclei, E(t) = f(t)E0sin(𝜔t) is the laser electric field, and f(t) is the envelope function. The expressions of He and HN are shown as follow: The three terms in Eq. (2) represent the kinetic energy, the electron-proton attraction and the electron-electron repulsion successively. The Coulomb softening parameters c, d are chosen to reproduce faithfully the first three electronic potentials of H2 (c = 0.7, d = 1.2375) [28, 32]. In Eq. (3), μ is the reduced mass of two nuclei. The TDSE is solved using a split operator spectral method [34] for the driving pulse of wavelength λ = 800 nm. The f(t) is chosen as the trapezoid envelope that rises linearly from zero to one during the first one optical cycle, afterward holds constant for eight optical cycles, and decreases linearly from one to zero during the last one optical cycle. This particular choice of f(t) cannot affect the nature of physical reality [35].The calculated ionization probability as a function of the time is shown in Fig. 4(a). One can see that not only the single ionization probability but also the double ionization probability with nuclear motion are much bigger than the case with fixed nuclei approximation. Because the ionization rate of electron is very sensitive to the internuclear distance, the effect of nuclear motion can explain well the rapid enhancement of the single ionization and the double ionization. The calculated result is consist with the fact that the change of the medium refractive index induced by the enhanced ionization induced by the nuclear motion leads to the obvious blueshift of harmonic spectrum at about 0-mm focus position in our experiment. The calculated harmonic yields with nuclear motion and fixed nuclei approximation are shown in Fig. 4(b). The harmonic intensity with nuclear motion is greatly suppressed by about 20 times comparing to the harmonic intensity with fixed nuclei under single molecular response. The calculated results are consistent with our experimental results: the harmonic intensity of H2 is dozens of times smaller than the harmonic intensity of N2 and Ar under the same conditions. We can conclude that both the single molecular response and the variation of phase matching condition suppress the harmonic intensity, with nuclear motion. In addition, the calculated harmonic spectrum from H2 with nuclear motion shows obvious spectrum broadening especially for the red side, comparing to the case of fixed nuclei. The calculated results give us supports to our explanation that the non-ignorable nuclear motion leads to the obvious spectrum broadening and the suppression of harmonic intensity, although the 1D TDSE calculations always overestimate the contribution of the long trajectories because the spreading of the continuum electron wave packet is neglected in the calculation.
Fig. 4 (a) Calculated ionization probability as a function of time with the TDSE simulation. (b) Calculated harmonic intensity with nuclear motion and fixed nuclei approximation.
4. Conclusions
In conclusion, we have experimentally observed an obvious spectrum broadening from the H2 harmonic spectrum. The comparison of the experimental result of H2 with the calculated one allows us to address a conclusion that the nuclear motion can lead to the spectrum broadening and decrease the harmonic intensity. The redshift induced by the asymmetry of ionization signals with nuclear motion and the blueshift induced by the enhanced ionization induced by the nuclear motion can result in the observed spectrum broadening of H2 molecular harmonic.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 11127901, 61221064, 11134010, 11227902, 11222439, 11404356, 61108012, and 11474223), the Zhejiang Provincial Natural Science Foundation of China (No. LY14F050008), Science and Technology Commission of Shanghai Municipality Yangfan Project (No. 14YF1406000), and Youth Innovation Promotion Association, CAS.
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